Title: Radiative transfer in circumstellar dust
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Title: Radiative transfer in circumstellar dust
Physical Description: vi, 90 leaves. : illus ; 28 cm.
Language: English
Creator: Harvel, Christopher Alvin, 1944-
Publication Date: 1974
Copyright Date: 1974
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Subject: Transport theory   ( lcsh )
Radiation measurement   ( lcsh )
Stars -- Radiation   ( lcsh )
Astronomy thesis Ph. D   ( lcsh )
Dissertations, Academic -- Astronomy -- UF   ( lcsh )
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
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Thesis: Thesis -- University of Florida.
Bibliography: Bibliography: leaves 87-89.
General Note: Typescript.
General Note: Vita.
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Bibliographic ID: UF00098170
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000582471
oclc - 14114730
notis - ADB0846

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RADIATIVE TRANSFER IN CIRCUMSTELLAR DUST


By

CHRISTOPHER ALVIN HARVEL















A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY










UNIVERSITY OF FLORIDA

1974

















































to my parents


____- ___ j
















ACKNOWLEDGMENTS


I would like to thank Professor Sabatino Sofia for

suggesting this problem to me and for the inspiration and

guidance he has given me toward a solution. I would also

like to thank Professors Frank Bradshaw Wood, Kwan-Yu Chen,

Howard L. Cohen, and James W. Dufty for serving on my com-

mittee.

I have enjoyed, greatly appreciated, and profited

by many valuable discussions with Andy Endal, Wayne McClain,

Joe Mullen, Liz Mullen, Al Rust, and Wilbur Schneider.

The curve plotting done by Andy Endal, the drafting

done by Charley Bottoms and Joe Mullen, and. the typing of

the entire manuscript done by Nadia Scheffer are also

greatly appreciated.

Computing time for the project was donated by the

Northeast Regional Data Center of the University of Florida

and the Central Florida Regional Data Center of the

University of South Florida.


___

















TABLE OF CONTENTS


Section Page

ACKNOWLEDGMENTS . . . . . . . . . iii

ABSTRACT . . . . . . . . . . . v

I. INTRODUCTION . . . . . . . . 1

II. BASIC RELATIONS . . . . . . .. 30

III. NUMERICAL PROCEDURE . . . . . ... 47

IV. RESULTS OF MODEL CALCULATIONS . . . . 56

LIST OF REFERENCES . . . . . . . .. 87

BIOGRAPHICAL SKETCH . . . . . . ... 90











Abstract of Dissertation Presented to the Graduate
Council of the University of Florida in Partial
Fulfillment of the Requirements for the
Degree of Doctor of Philosophy


RADIATIVE TRANSFER IN CIRCUMSTELLAR DUST

By

Christopher Alvin Harvel

August, 1974

Chairman: Frank Bradshaw Wood
Major Department: Astronomy

A procedure is developed to calculate the radiation

field as a function of position, direction, and wavelength

within a spherically symmetric circumstellar dust shell.

The dust shell is assumed to consist of grey isotropically

scattering dust particles in thermal equilibrium with the

radiation field and to be characterized by seven parameters--

(1) the radius of the central star, (2) the inner radius of

the shell, (3) the outer radius of the shell, (4) the total

optical depth of the shell, (5) an index which specifies

the density distribution, (6) the albedo of the dust parti-

cles, and (7) the temperature of the central star.

The temperature distribution and the radiation field

within several model dust shells are determined, and used to

calculate for each shell the spectral-energy distribution of

the radiation emitted. These models are compared to those

of other authors. It is found that fitting the visual

spectral-energy distribution of an observed object does not

necessarily mean that the model parameters so obtained are










consistent with the infrared part of the objects'spectral

energy distribution, and that the shape of the spectral

energy distribution can be strongly dependent on the index

in the assumed power law density distribution.

The procedure is applied to the two infrared objects--

HD 45677 and R Monocerotis.















SECTION I
INTRODUCTION


This paper will consider the problem of radiative

transfer in a spherical shell of dust particles illuminated

by a central star. In such a shell a dust particle will

absorb, reemit and scatter the radiation from both the cen-

tral star and the other particles of the shell. Since the

star will, in general, be much hotter than the particles,

the radiation field will consist of a visual part, due pri-

marily to the star and an infrared part due primarily to

the dust. Therefore, the solution of the transfer problem

must take into account absorption, reemission and scattering

of both these parts of the radiation field.

The solution of this transfer problem will yield the

quantities which characterize the radiation field at any

point in the shell and at any frequency the intensity and

the flux for example. Using these quantities we can de-

scribe the flow of energy from the star to the outer edge

of the shell and predict the observational characteristics

of the shell. Also, from a theoretical point of view the

radiation field within the dust shell is of considerable

interest, since it determines a number of physical quanti-

ties related to the matter in the shell. The temperature

of the dust particles, the radiation pressure on the dust









and gas and the excitation and ionization states of the gas

are all dependent on the radiation field.

From a knowledge of the dust temperature within the

shell we can limit the number of possible materials of which

the dust particles can be composed. According to Gilman

(1969), at temperatures above 15000K, refractory silicates

such as Al2SiO5 will be the most important condensates

around oxygen-rich stars while carbon will dominate around

carbon-rich stars below this temperature a number of other

compounds can begin to condense. A knowledge of the radia-

tion pressure as a function of distance from the star will

allow us to compare its effect on the material there with

that of other forces such as the gravitational force and the

force due to high velocity particles which might be ejected

from the central star. The ionization and excitation state

of the gas associated with a dust shell can be of consider-

able importance in calculating models for collapsing, rotat-

ing photostars interacting with an external magnetic field

(Prentice and ter Haar, 1971).

The observational characteristics of the shell which

the:solution of the transfer problem would yield are of

considerable interest because of the recent availability of

a great deal of observational information at infrared wave-

lengths. In the past few years a large number of objects

have been discovered which emit more infrared radiation

than expected. Many of these infrared objects emit most of

their radiation in the region to the red of 1i and have









gone undetected in the past because of the low sensitivity

of detectors in that region. Since 1965 the region between

1 and 20p has been surveyed extensively and several thousand

infrared sources have been discovered. A partial list of

these surveys follows: the California Institute of Technology

survey at 2.2p which discovered approximately 5600 sources

(Neugebauer and Leighton, 1969), surveys for red stars using

IN plates by Hetzler (1937), Chavira (1967), Ackermann et al.

(1968), and Ackermann (1970), and surveys at wavelengths

longer than 10P (e.g., Low 1970).

Many of the objects discovered in these surveys have

an excess of radiation within a given infrared wavelength

interval, as compared with a black-body with an effective

temperature appropriate to the objects observed spectral

type. This excess of infrared radiation is usually referred

to as an "infrared excess." Most of the 5600 objects dis-

covered in the 2.2p survey have been identified with late-

type giant stars (Grasdalen and Gaustad, 1971) but about 50

of the reddest of these objects could not be identified

with cataloged visual sources. These very red sources are

manifestations of a wide variety of physical objects, in-

cluding according to Neugebauer et al. (1971), HII regions,

planetary nebulae, M supergiant stars, late-type giant

stars, RV Tauri stars, novae, Be stars, and T Tauri stars

and related objects. Most of the infrared excesses asso-

ciated with the T Tauri stars and related objects are as-

sumed to result from thermal radiation from a circumstellar









dust shell. This assumption is also made for Be stars but

in their case it has been argued by some authors, e.g.

Woolf et al. (1970), that the observed infrared emission is

consistent with free-free emission from ionized hydrogen.

For those objects other than the Be stars and the T Tauri

stars and related objects thermal radiation by dust is prob-

ably present but may not be the principal cause of the

observed infrared excess.

Since a general solution of the transfer problem for

circumstellar dust shells would be readily applicable to the

T Tauri stars and related objects a more detailed descrip-

tion of these objects will be given at this point.

The T Tauri stars and related objects form an especial-

ly interesting group from both an observational and a

theoretical point of view. Poveda (1965a) pointed out in

1965 that T Tauri stars, which for the last 20 years have

been assumed to be young gravitationally contracting objects,

might have thick circumstellar dust shells. He argued that

such a shell would be a remnant of the contracting cloud

from which the star formed and that it would be bright in

the infrared. Shortly after this Mendoza (1966) found that

most T Tauri stars do have infrared excesses and later

Mendoza (1968) found that the bolometric luminosity of these

objects was from 1.3 to 6.6 times that expected from the

visual observations. For R Monocerotis, an object closely

related to the T Tauri stars, Mendoza (1968) found the

bolometric luminosity to be 58 times that expected from the









visual data. Figure 9 shows the spectral-energy distribu-

tion of R Monocerotis and it can be seen that most of the

observed luminosity is emitted beyond li.

The T Tauri stars are Ha emission stars, and in the

early 1940's all such emission stars were studied spectro-

scopically because of the intrinsic value of their spectra.

It was not until 1945 that Joy (1945) recognized T Tauri

stars as a distinct class of emission-line variables asso-

ciated with nebulosity.

From a study of eleven stars, Joy established the cri-

terion to be used in assigning stars to this new class. He

describes in detail each of the eleven stars and says that

the spectra in general are F5-G5, but that there is a small

variation in spectral type with the brightness. Also,

absorption lines are usually lacking but many bright lines

of low excitation are present. He points out the similarity

between the spectra of these stars and the solar chromo-

sphere. For five of the stars he finds color excesses, which

he attributes to strong selective absorption of the nebulosi-

ty. He finds that usually the emission lines seen are dis-

placed to the violet of the absorption lines (if any absorp-

tion lines are present), and that it is very difficult to

say anymore about the radial velocities of these stars.

In a review article on T Tauri stars and related

objects, Herbig (1960) points out that the only conclusive

test for membership in the T Tauri class is provided by the

spectroscopic data.









The classification system in use today is described in

and employed by the Third Edition of The General Catalogue

of Variable Stars (Kukarkin et al., 1969). The criteria

given in the catalogue are those adopted by the IAU Commis-

sion 27. These criteria are spectroscopic with one exception,

which is, that if the variable is connected with diffuse

nebulae it is designated InT whereas if it is not it is

designated IT. The prototype is stated to be T Tauri.

The catalogue breaks the irregular nebular variables

into several classes, one of which, InT, represents the T

Tauri stars. Its class Is is the same as Hoffmeister's RW

Aur class, and differs little from the InT class. Three

more of the classes defined by the catalogue, In, Ins, and

Inb, contain young objects of spectral class F-M with ir-

regular light variations (these have been called the Orion

variables). Of its nebular variable class the catalogue

class Ina (early spectral type Orion variables) differs the

most from class InT.

In some of the current literature stars falling into

all these classes are referred to as T Tauri stars and

related objects, although usually only in the sense that

the star is in an early evolutionary stage can it be grouped

with the T Tauri variables.

R Mon, the unusual object observed by Mendoza, is not

included in class InT because it is assigned by a Russian

author to spectral class A-Fep, which is too early a

spectral class for an InT variable. Joy (1945) originally









classified it as approximately G, and Mendoza (1968) classi-

fies it as K. Thus this star has been assigned spectral

classifications from A through K. This points out an impor-

tant fact; namely, the spectral classification of T Tauri

stars is very difficult because of the lack of a strong

absorption spectrum.

Most of these stars are rather faint; the brightest of

the class is about my = + 9 at maximum light. The magnitude

usually quoted for T Tauri variables is the mean of the

observed magnitude, even though this may not be the most

meaningful value.

For the last twenty years T Tauri stars have been

assumed to be pre-main-sequence (PMS) stars and recently PMS

models have become available which might help to verify this

assumption. The best proto-star models at present are those

of Hayashi (1966) and Larson (1969b). Both of these models

are directly applicable to the problem of the T Tauri

phenomena.

For his model Larson assumes that the initial proto-

stellar cloud is spherical, not rotating, and free of

magnetic fields. He does not assume, as Hayashi does, that

the collapse is homologous and that the density follows a

polytropic law. Initially the cloud has the Jeans radius

(R = vs[Gp]-1/2, where vs = the velocity of sound, G = the

gravitational constant, p = the density), and a density of

1.10 X 10-19 gm/cm3. The time scale for collapse in this

model goes as 1//p, so that the density gradient increases









with time, resulting in a dense core, which finally gains the

size and mass of a star. The core then evolves toward the

Main Sequence (MS) while accreting matter from the lower den-

sity cloud about it.

If the mass of the cloud is 2.5M.-or less, the proto-

star will become visible somewhere along the lower part of

the fully convective Hayashi track; if it is somewhat more

massive it becomes visible at some point on the radiative

track running horizontally toward the MS; if it is very mas-

sive (as massive as an 0 or B MS star) it does not become

visible until reaching the MS.

The time required in Larson's model for the accretion

of the envelope depends on the mass of the proto-star, and

ranges from 105 to 106 years.

Hayashi's computations start with a proto-star of much

higher density, and in his model the accretion of the cloud,

which brings the proto-star to the beginning of the fully

convective phase, takes place in a few decades.

The two models differ in their treatment of the shock

phenomena at the interface between the shell and the dense

core, but the latter parts of their evolutionary tracks are

in agreement. In both models we have a dense core and a

circumstellar shell, and Strom (1972) points out that we

should expect some young stellar objects to show evidence of

the remnants of their shells. These proto-star models

neither explain nor predict the variability, line emission,

and mass ejection observed for the T Tauri stars.









There are a number of reasons for believing that the T

Tauri stars are PMS objects: (1) their large red and infrared

excesses and deficiencies; (2) their position on the H-R

diagram (above and below the MS); (3) their association with

dense interstellar matter; (4) their association with O and

B stars; (5) their apparent rotational velocities; (6) their

anomalous Li abundance; (7) their polarization.

As noted earlier, Joy (1945) found that almost half of

his T Tauri stars had color excesses. Since then other

authors have found large infrared excesses for some stars,

and infrared deficiencies for others.

Mendoza (1966) did photometry in the UBVRIJKLM bands

from .361 to 51 for 26 stars, including T Tauri, RW Aur, R

Mon and Lk Ha 120. His observations show red excesses for

most of the stars, a deficiency (EV-R 0.2) for DF Tau most

likely caused by the Ha line in emission in the V band, and

infrared excesses (from 0.9 to 5V) for all the stars

(EV-M = 8.5 for R Mon).

Two years later Mendoza (1968) published another set

of photometric observations on Johnson's nine color system;

this time for 33 stars, some of which he had observed

previously (Mendoza, 1966). His results were similar to

those of 1966. He states that the two stars with the largest

infrared excesses are R Mon and R CrA. Most of the objects

observed by Mendoza (1966, 1968) were T Tauri stars.

In making the observations described above Mendoza

used the 60" infrared telescope at the Catalina observatory









in Arizona. He notes that making observations of these

objects is made difficult by their faintness and their

close association with nebulosity. He tried various dia-

phram sizes from 36 to 13 seconds of arc in diameter, and

found that in many cases a diaphram smaller than 13 seconds

was actually needed.

From his observed fluxes, Mendoza (1968) computes

bolometric corrections, and then absolute magnitudes for

several of these objects. He finds that they have high lu-

minosities, and speculates that Joy (1945) classified them

as dwarfs because of their rather wide absorption lines.

To explain his 1966 observations Mendoza (1966)

advanced two theories: (1) a multiple star system in which

one of the components is a very cool infrared star, and (2)

a core-envelope model in which the UBV photometry pertains

to the core and the infrared photometry to the envelope. In

this paper (Mendoza, 1966) he does not favor one theory over

the other.

In the second paper Mendoza (1968) states that his

previous core-envelope model is probably the explanation for

the large infrared excesses observed for objects such as R

Mon and R CrA. For the objects with small infrared excesses

(infrared just a few times the visual) he suggests that the

cause is heating of the stellar wind by cosmic rays.

He points out that his observations indicate that the

envelope around the core behaves like a neutral filter in

the visual (UBV) region. Assuming that all the infrared









radiation comes from this envelope, he finds that it would

have a radius of about 10 AU and a mass of about 10-3 M

Low and Smith (1966) published the results of their

photometry of R Mon at 20p. They found that at that wave-

length the star was about one fourth as bright as Mondoza

(1966) found it to be at 3.4p.

To explain their observations they computed the

spectral-energy-distribution to be expected from an optical-

ly thin circumstellar dust shell in which scattering and

reabsorption of infrared radiation emitted by the dust can

be neglected. The theoretical distribution they obtained

matched the available observations fairly well.

Strom et al. (1972) observed 42 stars in NGC 2264 at

1.6, 2.2, and 3.4p,and found infrared excesses and some

infrared deficiencies. The authors state that these obser-

vations are an indication that the stars have circumstellar

shells. They used profiles of the observed hydrogen lines

to estimate the surface gravities of some of the PMS A stars

in their sample of 42 NGC 2264 stars. From these surface

gravities they computed luminosities, and found that those

stars that seemed the faintest, as compared with their

computed luminosities, all had infrared excesses. They sug-

gest that these infrared excesses are due to a dust shell,

radiating in the infrared, energy it has absorbed from the

star in the visual part of the spectrum. The stars that

they found to have the largest infrared excesses, were those

that lie below the MS.









Strom et al. (1972) also found that these shells have

a very high ratio of total to selective absorption, which

would suggest large dust particles. The dust shells of

lowest temperature were around the latest stars.

They speculate that the infrared deficiencies observed

could be due to a very cool dust shell at a great distance

from the central star. Of course, observations in the far

infrared should find the missing radiation.

Martin Cohen (1973b)has observed stars in NGC 2264, IC

5146, VI Cyg and NGC 7000 (The North American Nebulae). NGC

7000 contains the T Tauri star Lk Ha 190, which was discover-

ed by Herbig in 1958, and has recently brightened visually

by 6 magnitudes.

Cohen used the 60" Catalina infrared telescope, and

made observations in all wavelengths in a single night (this

is very important since these objects vary rapidly). He

concluded that for most of these stars (many of them not T

Tauri stars) thermal emission from circumstellar dust is the

cause of the infrared excesses he observed. In a few cases,

he says that free-free emission from a circumstellar shell

may be important out to about 11 .

Breger (1972), like Cohen, finds evidence for circum-

stellar shells about PMS stars in NGC 2264. As well as

infrared excesses, he finds light variability, which he at-

tributes to circumstellar shells.

Strom et al. (1971) found a small amount of continuous

and Ha emission, similar to that found for T Tauri stars, in









the spectra of some A-F stars in young clusters. They attri-

bute this to very thin circumstellar shells. They also say

that the strength of these features increases as the

circumstellar absorption does.

According to Strom (1972) we should expect to see free-

free (electron-H ion) and free-bound emission from envelopes

in the infrared, if they have sufficient size and density.

Dust envelopes, as stated previously, would show thermal

infrared reradiation; thus, Strom (1972) ascribes the cir-

cumstellar emission in the optical region of T Tauri spectra

to gas (this is the emission that washes out the normal

absorption spectrum). He says that, in a joint paper, Kuhi

and the Stroms indicate that gas shell emission may also do-

minate the infrared region.

In support of his idea that gas and not dust shell

emission is dominant in the infrared, Strom (1972) offers

the following argument and observational evidence. He

finds that for some young stars (especially A-F stars) with

circumstellar shells, Ha emission and infrared excesses

there exists a correlation between log (Ha intensity) and

the L magnitude (3.4p). In an optically thin case of the

dust model the dust emission is proportional to the optical

depth (T = p x length), whereas the gas emission (Ha)

depends on p2 x length. Therefore, unless we believe that

all circumstellar shells have the same densities, the dust

theory is difficult to defend, because we would not get the

observed correlation.









According to Herbig, in the Taurus cloud, Orion

nebulae, NGC 2264, NGC 6530, and IC 5146, we find that if we

fit the brightest T Tauri stars to a MS with a distance modu-

lus of 6 the Ml to M3 stars observed are too bright. Thus,

the observed MS in these regions appears to be turned up on

the red end; however, the observed MS does not curve upward

more and more as we observe later and later stars, but runs

parallel to the Zero Age Main Sequence (ZAMS) after sharply

breaking off from it at about AO. This observed MS forms a

band with its upper edge several magnitudes above the ZAMS.

Herbig (1960) believes that the uncertainties in the

observations and theory are large enough so that we need not

be concerned by the fact that the observed MS does not curve

upward. He also believes that part of the spread in the

observed lower MS is caused by a spread in the times of

stellar formation in the cluster.

Hayashi's evolutionary tracks for proto-stars would

lead us to expect young stars to lie above the ZAMS.

Poveda (1965a) first brought attention to objects

falling below the ZAMS in NGC 2264. He said that their lo-

cation below the ZAMS was caused by thick circumstellar

dust shells.

Strom et al. (1972), as we have seen, were able to

show that those stars, in NGC 2264, lying below the MS would

be above it, if their circumstellar shells were removed.

Presumably, the stars below the MS are younger than

those above it, and have not had time to flatten their









circumstellar shells. According to Mendoza (1968), the T

Tauri stars with the least infrared excess are the oldest,

and have the most flattened disk-like shells. If we are

looking at a disk edge on, we should see a large discrepan-

cy between the luminosity as determined from the spectral

classification, and the bolometric luminosity observed.

Strom (1972) finds, for a sample of 10 Herbig Ae and Be

stars, that for three, Lgp > > Lbolometric

According to Strom et al. (1972), the presence of

late-type stars near the MS in young clusters is caused by

circumstellar obscuration, and not by a large spread

(several x 107 years) in times of stellar formation, as

argued by Herbig (1960).

T Tauri stars are found directly associated with the

clouds in which they presumably formed. They can not be

interlopers from the general field around the region in

which they are found, since their space density averages 1

or 2 orders of magnitude above that of the field stars of

the same luminosity (Herbig, 1960).

Cohen (1973c) has observed the T Tauri type objects

at the tips of several cometary nebulae. Mendoza (1966)

states that T Tauri and R Mon illuminate bright nebulosities.

Both of these nebulosities are variable cometary nebulae.

R Mon is in NGC 2261, Hubbles variable nebulae, and T Tauri

is in NGC 1555, Hind's nebulae.

Poveda (1965c) describes the cometary nebulae

(elephant trunks, bright rims, etc.) as HII regions expand-


II









ing into a surrounding HI region, or as a HII region expand-

ing past an area of high density, compressing and deforming

it. If either of these explanations is correct, the tip of

such a nebulae will be a region of high density, suitable

for star formation. Thus, cometary nebulae are very young

phenomenon illuminated by even younger stars.

If we assume the stars have the space velocities of

the eddy of gases from which they formed (about 2 km/sec),

and if we estimate the size in km of the tip of the nebulae,

we can estimate how long it would take the star to drift

out of the tip, and thereby arrive at an upper estimate for

the age of the star. Poveda does this for several nebulae,

and finds ages running from 5 x 103 to 50 x 103 years.

Mendoza (1968) also makes a calculation to obtain the

ages of some T Tauri stars. He uses the PMS evolutionary

tracks of Iben to determine the age of a star from its

position on the H-R diagram. The ages Mendoza obtains in

this way agree with those of Poveda for some stars and

disagree for others.

For R Mon the two methods agree rather well, but

for the other stars Poveda's results are, in general, lower

than Mendoza's.

Herbig (1960)states that in regions such as Orion, we

find T Tauri stars associated with 0 and B stars, which are

considered to be young objects. It seems logical then, to

consider the T Tauri stars to be of similar age (hence

young), but of lower mass, and thus not as greatly evolved.









Joy (1945) classified the T Tauri stars as MS dwarfs,

which we now know to be incorrect. Those T Tauri stars

that do show absorption lines, have very wide lines, which

might indicate that they were luminosity class V objects.

However, it is now believed that these abnormally wide

absorption lines are caused by rotational broadening (Herbig,

1960).

The rotational velocities found for these stars

(20 km.sec < v sin i < 65 km/sec) are usually high for

stars of their spectral class. Normal G-K stars have

v sin i < 15 km/sec.

If we compute the radii of the T Tauri stars from

their observed luminosity and color-temperature, and then as-

sume they contract without loss of angular momentum along a

Hayashi track, we can find their rotational velocities on the

MS. When this is done, the computed values are in close

agreement with observed values of v sin i for MS stars of

comparable mass. It may of course be true that the line

broadening is not caused by rotation.

The T Tauri stars as a group appear to be overabundant

in Li, as compared with the Sun, by 1 or 2 orders of magni-

tude, which is taken as an indication of their youth (Herbig,

1960). Herbig compared the Li abundance of the T Tauri

stars to that of chondritic meteorites, which presumably

have the Li abundance of the pre-solar nebulae.

It is possible that the large abundance of Li is being

caused by Li production in the photospheric layers of the T









Tauri stars, in which case it may not be an indication of

their youth. It seems unlikely though that such Li produc-

tion is actually occurring, since the most plausible

mechanisms suggested for Li production require large, strong

magnetic fields, which have not been observed.

Zellner (1970), Capps and Dyck (1972) and Breger and

Dyck (1972) observe polarization for some stars in the near

and far infrared. Zellner's measurements for R Mon indicate

polarization typical of large grains (radius u 0.1l) while

the other authors cited indicate that their measurements

favor electron scattering similar to that observed for Be

stars. In a more recent paper Zellner and Serkowski (1972)

state that R CrA exhibits polarization similar to R Mon.

They suggest that in objects such as R Mon and R CrA the

dust cloud producing the infrared excess is in the form of

an equatorial disk and the light seen at shorter wavelengths

is scattered light which has filtered out the poles of this

disk.

Breger and Dyck find that between 3000 and 7500 A,

four out of a sample of 35 stars in NGC 2264 show intrinsic

polarization. Most of the stars they observed were not T

Tauri stars, but of the four showing polarization one is a

T Tauri star. Breger and Dyck comment that its polarization

has a different X dependence from that they observe for the

other three polarized stars in their sample, and that its

polarization is also different from that of T Tauri, as

given by Serkowski (1971).









These investigations indicated the presence of circum-

stellar shells about T Tauri stars and stars in clusters as-

sociated with T Tauri stars, such as NGC 2264.

Some other important characteristics exhibited by

young stars and T Tauri stars are the following: (1) mass

loss and the P Cyg effect, (2) mass accretion and the inverse

P Cyg effect, (3) ultraviolet excess, and (4) correlations

of the visual magnitude and the infrared emission, of the Ha

UV and IR emissions, and between mass infall and the V

magnitude.

Most T Tauri stars show a P Cyg type spectrum, that is,

their emission lines havebleshifted absorption features

superimposed. According to Strom (1972) and Herbig (1960),

the theory that this effect is caused by an expanding envel-

op with self absorption is widely believed.

Herbig (1960) states that the star T Tauri has a set

of two absorption lines superimposed on its H and Ca emis-

sion lines, corresponding to radial velocities of -70 and

-170 km/sec, and that RY Tau has one line corresponding to a

radial velocity of -90 km/sec.

From an analysis of the forbidden-line radiation, com-

ing from the small forbidden-line region surrounding T Tauri,

Herbig says that Varsavsky, in 1960, finds a mass loss of

about .5 x 10-5 Me/yr for T Tauri, assuming that the

forbidden-line region is 750 AU in diameter, and that the

region is sustained by material rising at 170 km/sec.

The accretion of matter by some young stars is indicated










by the presence of an inverse- P Cyg spectrum (sometimes

called a YY Orionis spectrum). Out of a sample of 23 UV-

excess stars observed by Walker (1969) in NGC 2264, he finds

10 show an inverse P Cyg effect. None of these are T Tauri

stars, but they are all closely related to T Tauri stars. In

1972, Walker (1972) reports finding two more young stars

with inverse P Cyg effect, RR Tauri and 01C Orionis.

T Tauri spectra show a strong continuum in the visual,

which is sometimes called the 'blue continuum', and a

continuum starting in the XX3700-3800 region and rapidly

rising toward the shorter wavelengths. Unlike the blue

continuum, this UV continuum adds an appreciable amount of

energy to the luminosity of the star. According to Herbig

(1960), this UV continuum has been attributed to synchrotron

emission like that from some solar grains, and to the running

together of the higher Balmer lines. The latter of these

hypotheses he refers to as B6hm's hypothesis. The higher

Balmer lines blend together because of turbulent broadening,

according to this hypothesis.

Mendoza (1966) finds UV excesses for the majority of

the 26 stars he observes. Anderson and Kuhi (1969) give

detailed observations of the T Tauri star AS 209 (1900: 16h

43m.6, -140 13'), and report that it shows a large UV excess.

They say they favor B6hm's hypothesis, since Kuhi's

observed line width for the HB line, indicates a turbulent

velocity as large as 100 km/sec, which would be more than

enough to blend the higher Balmer lines.









In general, Ha emission and UV excess in T Tauri stars

is correlated, but for individual stars there does not appear

to be a correlation between the two as the star varies in

brightness. In the case of AS 209, Anderson and Kuhi looked

for such a correlation, and found none.

Various correlations have been noted between observed

T Tauri phenomena. Herbig finds that as RW Aur gets fainter

in the V band, it gets redder. Anderson and Kuhi (1969) find

that AS 209 increases its UV excess as it gets fainter in the

V band. Cohen (1973a) finds that as T Tauri gets fainter in

the visual it gets brighter in the near infrared.

Walker (1969) does a detailed study of SU Ori, an Inb

star with inverse P Cyg spectrum and UV excess, and finds

that when the redward displaced absorption feature is most

prominent, the star is brightest. He shows five spectro-

grams of SU Ori taken when the star was at different magni-

tude, and the redward displaced absorption line disappears

as the star gets dimmer.

Poveda (1965b, 1965c) suggests that the circumstellar

shells about T Tauri stars and related objects are pre-

planetary systems. He says that some of the variability of

these stars must be due to eclipses by proto-planets (large

clouds along the orbits of the planets-to-be), and that

phenomena such as FU Orionis occur because the circumstellar

material can form into large particles very rapidly (a few

years); thus, rapidly dropping the optical depth of the

material, and brightening the star.









In light of the observed characteristics of the T

Tauri stars and related objects described above it is not

surprising that there have been a number of papers published

in the past few years dealing with the problem of radiative

transfer in circumstellar envelopes. In each of these

papers the author makes a number of assumptions about the

object or objects he is attempting to model in order to sim-

plify the radiative transfer problem. These simplifying as-

sumptions tend to reduce the accuracy and the versatility of

the resulting model and should therefore be avoided if

possible.

In all of these papers it has been assumed that the

envelope is spherically symmetric about the central star and

that the effects of the spherical symmetry are significant.

If the entire shell were metrically thin and distant from

the central star it would be possible to simplify the prob-

lem by assuming the shell to be stratified into parallel

planes. This simplification could also be made if the outer

layers of the envelope were optically thick. However, it

appears from the observations that the circumstellar clouds

about the T Tauri stars and related objects can not be

adequately represented by a model based on a plane parallel

approximation. The assumption of spherical symmetry while

much better than the simple plane parallel assumption is

still an idealization since it has been pointed out that

visually these objects appear to have a very complicated

outer structure (e.g., Herbig, 1968).









In all these papers it is assumed that a power law of

the form K(r) = [C/rn] governs either the volume absorption

coefficient or the number density of particles. Here r is

the distance from the central star, C is a constant, and n

is a variable index (Chandrasekhar, 1934). Some assumption

as to the functional relationship between K and r is neces-

sary (see Section II), and this one seems to allow reason-

able flexibility with the adjustment of only one parameter

(the constant is determined by the geometry and total

optical depth of the shell). In a few of these papers it is

further assumed that n = 0; that is, that K(r) is constant

with r

Several authors (Stein, 1966; Low and Smith, 1966;

Krishna Swamy, 1970) have considered the less general prob-

lem of radiative transfer in an optically thin circumstellar

shell in which scattering and reabsorption of infrared radia-

tion emitted by the dust can be neglected. Stein's paper

pertains to the observed infrared excesses of certain B

stars; Low and Smith's, to the much larger infrared excess

of R Monocerotis and Krishna Swamy's, to the infrared excesses

of several infrared objects, among them, VY Canis Majoris.

Stein in his paper states that the total optical depth

of the shells he is considering is about 10-3. For a shell

this optically thin it is quite reasonable to assume that

the star is the only significant energy source in the system.

In the paper by Swamy and in the the one by Low and Smith,

the objects modeled (e.g., VY CMa and R Mon) show an amount









of infrared radiation as compared with the optical which

indicates that these objects have optically thick shells

(Mendoza, 1966; Low et al., 1970).

Huang in one paper (Huang, 1969a) considers the two

extreme cases of very small and very large optical thickness

and then in a second paper (Huang, 1969b) considers the more

general problem of intermediate optical thickness. In both

these papers Huang makes the assumption that the opacity of

the dust is proportional to the geometrical cross section

of the grains.

In the first paper he solves the transfer problem for

an optically thin shell in much the same manner as Stein

(1966). His approach to the problem of the optically thick

shell is analogous to that of Chandrasekhar (1934) for

extended stellar atmospheres.

In the second paper he solves the transfer equation

for a spherical shell having its inner boundary in contact

with the surface of the central star, assuming grey

particles, the Eddington approximation, isotropic scattering

and Local Thermodynamic Equilibrium. He does not actually

state that the shell is in contact with the surface of the

star but that the radiation at the inner boundary of the

shell is completely diffuse. For all practical purposes as

long as the inner edge of the shell is within a few stellar

radii of the surface of the star the radiation field can be

considered diffuse.

Huang (1971) published a third paper dealing specifi-









cally with shells distant enough from their central stars

such that the stars could be considered point sources. He

points out in this paper that the assumption of a complete-

ly diffuse radiation field throughout the shell represents

an idealized case of one extreme while the assumption that

the star is a point source represents the other extreme.

In his second paper Huang (1969b) argues that the as-

sumption that the particles scatter radiation as grey-bodies

is valid. He states that in the case of interstellar red-

dening blue light is preferentially scattered out of the

line of sight, whereas in the case of a circumstellar dust

cloud we observe scattered radiation coming in all direc-

tions; such that, the amount of radiation of a certain wave-

length which is lost in direct transmission along the line

of sight is regained due to the overall scattering process.

For his intermediate case Huang (1969b) assumes that

the radiation field can be broken into two parts a visual

part and an infrared part. He then proceeds to solve the

overall transfer problem as though it were two less diffi-

cult problems: (1) a nonconservative scattering problem in-

volving only the stellar radiation in the visual (the

infrared emission of the star is assumed to be negligible)

and (2) the problem of an atmosphere in local thermo-

dynamical equilibrium for the infrared radiation emitted by

the dust, with an energy source, namely, the visual radia-

tion absorbed in the nonconservative optical region. There-

fore, the results for his intermediate case described in his









second and third papers (Huang, 1969b, 1971) refer to these

two broad wavelength regions.

In his results Huang (1971) states that the radiation

field and the temperature distribution do not critically

depend upon the index n in the expression assumed for the

density distribution and indirectly that the ratio of the

inner to the outer radii of the shell is an unimportant fac-

tor. In Section IV it is shown that these results are not

in general correct.

Huang does not apply this method to any observed

object, but in 1970 Herbig (1970) uses it successfully in

the development of a model for VY Canis Majoris, and recent-

ly Apruzese (1974) has used it to model the infrared object

HD 45677 (a Be star with a large infrared excess). The

approach followed by Apruzese parallels Huang's for the in-

termediate case and diffuse radiation except that he

modifies Huang's technique slightly to allow the determina-

tion of the radiation field in ten spectral regions to the

blue side of 1w For each of these spectral regions he

has in effect a problem of nonconservative scattering.

Apruzese's model will be considered further in Section IV.

Larson (1969b) in studying the emission of collapsing

protostars finds an approximate solution for the radiative

transfer in a spherical dust shell in which the particles

scatter none of the radiation but absorb an amount propor-

tional to A-P, where A is the wavelength and p is a varia-

ble index. He derives an approximate expression for the










temperature distribution in the cloud and using this solves

for the emitted spectrum. The weighting function integra-

tion method he develops to compute the emitted spectrum

from the known temperature distribution is also used by

Herbig in the paper mentioned above. For the index n which

is used to establish the density distribution in the cloud

Larson uses 1.5 and states that this is the value determined

by his dynamical calculations and published in a previous

paper (Larson, 1969a).

Hummer and Rybicki (1971) have a procedure involving

iteration on the Eddington factor, K/J, which they use to

solve the transfer problem, under the assumption of the con-

servative grey case, for a spherical cloud extending from a

point of radiation at its center out to some finite radius,

R.

None of the papers mentioned above adequately deal

with the problem of radiative transfer in circumstellar en-

velopes of intermediate optical depth. The only two proce-

dures that come close to solving this case are those due to

Huang (1969b, 1971), Larson (1969b), and Apruzese (1974).

They all assume that the density in the shell is proportion-

al to 1/rn and Huang and Apruzese assume that the particles

are isotropic scatters Larson neglects scattering. The

first of these assumptions seems as good as any other for

the density distribution and is not critical to the solution

of the problem. The second assumption is necessary in

order to take advantage of the spherical symmetry of the









shell. It allows us to reduce the problem to one involving

only two spacial coordinates. If the particles are assumed

to scatter according to an arbitrary phase function the

problem becomes much more complex. Huang assumes that the

opacity of the cloud is due to the geometrical cross section

of the particles and therefore that it is grey, whereas,

Apruzese and Larson assume that the opacity is wavelength

dependent. The procedure described by Huang is restricted

to cases where the central star can be considered a point

source or the radiation is completely diffuse. It is

further restricted to cases where the Eddington approxima-

tion is valid. Most importantly his procedure does not de-

scribe the radiation field at a particular wavelength but

merely breaks it into two regions visual and infrared. The

procedure described by Apruzese is based on Huang's proce-

dure and involves one major improvement his procedure can

be used to determine the radiation field at a number of

wavelengths in the visual. Larson's procedure, as well as

neglecting scattering, uses a rather crude approximation

for the temperature distribution in the shell. His proce-

dure does however allow the determination of the spectral-

energy distribution emergent from the shell in the visual

and infrared.

In Sections II and III a technique is developed

capable of solving the transfer problem for the radiation

field at any point in a circumstellar shell, for any wave-

length, and any optical depth.









This technique is applicable to cases where the

central star can be considered neither a point source nor a

diffuse source of radiation. To simplify the problem the

following assumptions are made: (1) the cloud is spherically

symmetric about the central star, (2) the cloud is made of

small particles which have no preferential orientation, (3)

the particles reemit absorbed radiation isotropically and

scatter radiation isotropically, (4) the particles have an

albedo which is independent of wavelength, (5) the particles

radiate as grey-bodies at a temperature T where T is a func-

tion of the radiation field about the particle, (6) the

volume absorption coefficient as a function of the distance

from the central star, K(r), is constant with wavelength and

proportional to the density p(r), and (7) the optical depth

T and the distance from the central star are related by the

usual expression (Chandrasekhar, 1934),

dt r
dT = b n t =


where b is a constant.

This technique characterizes each dust shell by a set

of parameters related to the size of the shell and the

density distribution within the shell, the size and tempera-

ture of the central star, and the albedo of the dust

particles within the shell. The equation of transfer is

solved in integral form by a computer code using an itera-

tive procedure.
















SECTION II
BASIC RELATIONS


In this section the physical and mathematical relations

necessary to solve the transfer equation for the radiation

field in a circumstellar dust cloud will be developed. These

relations will be in terms of a number of physical parameters

that characterize the dust cloud.

The parameters to be used can be placed into three

groups, one that specifies the geometry of the cloud, one that

specifies the albedo of the particles, and one that specifies

the temperature of the central star. The first group consists

of five parameters, the radius of the central star, the inner

radius of the cloud, the outer radius of the cloud, the total

optical depth of the cloud, and an index which specifies the

density distribution. These five quantities will be desig-

nated respectively as R,, RI, R, TI, and n. The second and

third groups have only one member each: the albedo of the

particles, designated o in the case of the second group and

the temperature of the central star, designated T*, in the

case of the third.

For some set of these seven parameters the dust cloud

model should be capable of reproducing the observed spectral-

energy distribution of an object such that that set of param-

eters can be considered to characterize the observed object.










Consistent with the simplifying assumptions set forth

in Section I, relations can be derived for a number of useful

quantities. Using the expression for dT given in Section I

we can derive the relation for the absorption per unit volume

at a distance r from the central star, K(r). We have


dr = Rdt


d bRn-ldr
rn

but

dT = K(r)dr,

and therefore,
bRn-1
K(r) =-
rn

or
B
K(r) = (1)
n
r

where B is a constant. We can also derive expressions for

the optical depth at r, T(r), and for B in terms of R R,

TI, and n. In the following n $ 1 unless specified otherwise.

We have
r
T(r) = f K(r)dr
o

or
r
T (r) = Bf r-ndr
0

and, therefore,
l-n
T (r) = B( r ) + C (2)
1-n


where C is a constant of integration. Since T(RI) = TI and











T(R) = 0 we have


1-n

and
Rl-n
0 = B ( -- ) + C
1-n

from which we obtain,

C = B ( R--n).
1-n

Substituting this into (3) we have for Ti,


R 1-n
TI = B( )
l1-n


R1-n
- B( )
1-n


B (R 1-n R1-n
I 1-n


So
I (l-n)
B =
(R 1-n RI- )


and we obtain from equation (1)


K(r) = (1-n)
(R1-n R 1-n)rn
I


if R < r < R


= 0, if r > R, or r < R

and from equation (2)


(r) = T(R1- rl-n)
Tir) = R-----
(R1-n R 1-n
I









In the case with constant density we have n = 1 and

the relations for K(r) and T(r) become

-1
K(r) = TI[Zn(R/RI)r] RI 5 r < R


= 0, r > R, r > R (7a)



T(r) = in(R/r) [ n(R/Ri)]- (7b)


Equations (6) and (7b) apply only for RI r < R. For

r < RI, T(r) = TI, and for r > R, T(r) = 0.

The spectral-energy distribution is observed in terms

of flux received at the surface of the earth per unit wave-

length, so this is the quantity we want to calculate with

our model. If we define F(A,r) as the flux at wavelenth A

through a unit area of the shell of radius r about the

central star, where F(A,r) is in units of ergs/cm2 -

str sec, we have


F(X,D) = F(X,R)R2/D

where D is the distance between the object and the earth.

Note that if D is not a known quantity we can still find a

scaled flux and thus determine at least the shape of the

spectral-energy distribution.

We also have

F(A,r) = f I(A,r,u)cos0 dw (8)


where I(A, r, w) is the specific intensity of radiation at

wavelength X, at any point a distance r from the central









star, and coming from some direction w radians from the cen-

tral star. The integration is over 47 steradians. Because

of our assumption of spherical symmetry we can define I

uniquely with reference to only two of the three spherical

coordinates, r and 6; there will be no 4 dependence. It is

also true that I(A, r, 6) = I(X, r, -6), so the integration

over 6 need only be carried out from 0 to i. Thus,

Tr
F(X,r) = 2 /f I(A,r,6)sin6 cosO d6
0

where 6 is an angle in a single plane.

In order to determine F(A,r) and the F(A,D) we must be

able to calculate I(A,r,6) numerically. This can be

expressed as

I(A,r,6) = m e-t S(X,re,6e)dt, (9)
0

where S(X,r ,e ) is the source function for radiation of

wavelength X emitted in a direction 6e radians from the

central star, at any point a distance re from the central

star. The integration is carried out over optical depth

along the line of sight, from a point at a distance r from

the central star and in a direction 6 radians from the cen-

tral star. The upper limit for the integration, Tm,

represents the maximum optical depth attainable along the

line of sight and will always be less than 2TI. The quan-

tities re and 8e are functions of r, 8 and t. Here t is

optical depth along the line of sight and the variable of

integration. Thus we have re = re(O,r,t) and e = e(6,r,t).

The expression for I(A,r,6), giveniin equation (9),









does not help us unless we can evaluate S(A,re,e ). The

source function can be expressed as the ratio of the emission

to the absorption coefficient, that is,


S(X,r ,ee) = j(X,r ee)/K(r ),


where j(X,re,8e) gives the emission per unit volume at a

distance re from the central star, at wavelength X, and in a

direction ee radians from the central star. Since the dust

model is based on the idea that the dust scatters part of the

radiation it intercepts and absorbs and reradiates the rest,

we can write S(A,re, e) in the following form:


S(X,re,Oe) = [jB(A,r, ee) + js(X,re,Oe)] / K(re) (10)


where k(A,re,9e) has been broken into two parts, scattered

emission and thermal or grey-body emission, designated

respectively js and jg .

It is evident that both emission terms in equation (10)

are related to the radiation field, I(X,re,6), at re. There-

fore, we need to determine the exact functional relationships

expressing jB(X,re,6e) and js(A,r e) in terms of I(X,re,6).

In doing this we will first need to derive expressions for

re = re(O,r,t) and 0e = 8e(8,r,t).

From consideration of the geometry in Figure 1, we

have

re(8,r,t) = (r2 + s2 2rs cosS)1/2, (11)

where s is the metric distance along the line of sight to the

point of emission and


C~









e = sin-1 [(r/re)sin8].


The variable of integration along the line of sight, t,

can be evaluated from the expression

s
t = f K(r) ds (12)
o

once we have assigned a value to n in the formula for K(r).

The amount of radiation scattered by a particle will

be that amount which is intercepted by the cross sectional

area of the particle and not absorbed; that is; the amount

intercepted times o The amount of this radiation scattered

in a direction De will, in general, be given by


js(X,re'e)=(kpo /4T) I P(a)I(X,re,O)du, (13a)

where kp is the cross sectional area of the particle and

P(a) is a phase function defined such that

(1/4T)/ P(a)dw = 1


The quantity a is the angle through which the radiation is

scattered and is thus a function of 0 at re and of e (see
e
Figure 1).

Since we are assuming isotropic scattering, where P(a)

= 1 for all a, we can drop ee from our notation and rewrite

equation (13a) as


js(X,re) = (kpwo/4T) I I(A,re,6) dw (13b)


Equation (13b) gives the radiation scattered per particle

and we want the radiation scattered per cm3. To derive the









needed form we must replace k in equation (13b) with the

cross sectional area of all the particles in a cm3, which we

will denote k. Then,


js(A,re) = (kwo/4r) f I(A,re,O)dw

which because of spherical symmetry we can rewrite as

7T
js(A,re) = (kwo/2) I I(A,re,6)sine dO. (14)
0

Two additional equations for js are formed by separat-

ing I(X,re,6) into a term due to the cloud and a term due to

the star. Let y(O) be defined such that y(6) = 1 if

0 < < *(re) and y(6) = 0 if 6 > 8*(re), where e*(re) =
-1 R.
sin (1 ) is the angular radius of the central star as seen
e
from a distance re; then


I(A,re ) = Ic(A,re,e) + y(6) I*(A)


In the above equation I the term due to the cloud, is given
c
by
TC t
Ic(A,re,9) = f et S(A,r) dt

where r = r(O,re,t) is obtained analogously to re (see equa-

tion 11),

c = Tm y() (Tb Tm)

and Tb represents the optical depth along the line of sight

to the inner boundary of the cloud. The term due to the star

is I.(A) = e-TbB(X,T*), where B(X, T*) gives the Planckian

emission of the star.









From equation (14) and the above expression for I we

have
7T
js(X,re) = (kow/2)[ / Ic(A,re,O)sine dO +
o
8*
f I,(A)sinO de ], (15a)
0

and if we are only interested in the bolometric emission, we

have
7T
js(re) =(kow/2)[ / I (re,6)sinO dO +
0
6* Te-T I oT
I e (- -)sin6 de ], (15b)
0


where



0
Ic(re,6) = / I(A,re,e)dX,and a (15c)
o

is the Stefan-Boltzmann constant.

When 6, is very small we can use the following

approximate versions of equations (15a) and (15b):

Tr
js(X,re) = (kmo/2) I Ic(X,re,O)sin6 dO
0


+ (kmo/4n)exp(Te-rI)R B(A,T,)/r2 (16a)


and

7T
js(re) = (kwo/2) I Ic(re,e)sin6 dO


+ (kw/4r)exp(e-T )R2 oaT/r2 (16b)


where T is the optical depth from the surface of the cloud

to re and the other quantities used have been previously

defined.









We now turn to the problem of determining the grey-

body emission of the particles, jB(A,re,6). As long as any

elemental volume is small enough for every particle in it

to experience nearly the same radiation field, the amount of

energy absorbed will be independent of the shape of the

element and its orientation. This absorbed energy will be a

function of I and the effective cross sectional area of the

particles per unit volume, k. Thus, we can derive an expres-

sion for k by considering the energy absorbed by an idealized

unit volume; namely a small cylinder of unit height and unit

cross sectional area oriented such that the incident radia-

tion is perpendicular to one end.

For such a cylinder the amount of the intensity lost

due to absorption, IL, will be IL = I(1 e ), where T here

is the total optical depth along the axis of the cylinder,

or if we assume the medium to be thin enough so that occulta-

tion of one particle by another are rare, we have

IL = (Apart/Acy)I, where Apart and Acy respectively denote

the cross sectional areas of the particles and the cylinder.

Now we can write

part = (1 e)Acy
Apart cy

and since T < < 1, we have

A = [1 (1 T)]A = AC T
part cy cy
or

Apart = AcyK(r) h

where h is a length over which we have an optical depth T and









K(r) has been assumed constant over this length. For a cyl-

inder of unit volume we can take Ay = 1 and h = 1; then

Apart/unit vol = K(r), and k(r) = K(r).

We now need to derive a relation between the effective

cross sectional area of the particles in a volume and the

effective surface area enclosing that volume. Let A be the

sum of the effective surface areas of all the individual

particles in a volume element V and let k be the sum of their

effective cross sectional areas. Define the effective cross

sectional area of a single particle, to be its geometrical

cross sectional area averaged over all possible orientations.

Now if the number of particles in V is large, and they have

no preferential orientation; then k will not vary with the

direction of observation w, and we will be able to derive a

relation for A in terms of k; thus defining A.

For the volume, V, the energy absorbed, Ea, is given

by the relation


Ea = I I(C)k d ,



if the density is low enough such that occultations of one

particle by another can be neglected. The energy emitted,

Ee, is given by Ee = OT A. Therefore,

f I(w)k dw = oT4A ,



if the particles are in thermal equilibrium with the radia-

tion field. This can be rewritten as









T4 = (k/Ao) f I() dw (17)


and if we let W = k/A, we have


T4 = (W/a) f I(w) dw (18)


This equation is true for any volume of particles and any

radiation field.

If we apply equation (18) to a volume situated in a

black-body cavity at temperature TB, we have


TB = (W/o) I B(TB) dw,


and since B(TB) is constant with w, we have

4
TB = [W47B(TB)]/a (19)


Since B(TB) is not a function of X, it represents the

Planckian integrated over A. Thus,


B(TB) = i B(X,TB)dX = (aT)/T ,
0

and equation (19) becomes T4 = 4WT4 from which we obtain
B B
W= 1/4, a value indentical to that obtained for a sphere.

Substituting this value for W into equation (18) gives

us
1/4
T =[(1/4a)f I(w)dw]/4


This equation can be used to find the temperature of any

grey-body due to an arbitrary radiation field, I(w),

provided that the body is tumbling randomly.

Since W is the ratio of the effective cross sectional









area to the effective surface area of a particle, A = 4kp

for any particle.

From our assumptions that the particles have no pref-

erential orientation and that they are in thermal equilib-

rium with the radiation field, we have that A = 4kp and

that

F(A,T) = (1 w )iB(B,T)


where A is the effective surface area of a particle, T

is the equilibrium temperature of the particle, B(A,T) is

the Planckian and F(A,T) is the flux per unit effective

area of the particle. The thermal isotropic emission of a

single particle can now be written as


jB(X) = (A/4T)F(X,T)

or

jB(x) = k (l-wo)B(A,T)

The corresponding emission for a unit volume of such

particles will be


jB(A,re) = K(re) (-W )B(A,T)


Note that the temperature T, like K, is a function of the

distance from the central star, r.

If this emission is considered only bolometrically,


F(T) = (1-o) aT4









jB(r ) = [K(re)/F] (l-Wo)oT4 .(20)


The condition of thermal equilibrium gives us the

relation

Ea(r) = Ee(r) (21)


where E (r) and E (r) denote,respectively,the energy absorbed
a e
and emitted by a unit volume of particles at a distance r

from the central star. Now from equations (20) and (21) we

have

Ea(r) = I (1-oo) (0/)K(r)T4 (r)dw ,


and since the emission is isotropic


Ea(r) = 4K(r) (l-w )oT4(r)


From this formula for E we can obtain the following

expression for the temperature,


T(r) = {[1/4a(l-mo)] [E (r)/K(r)]}1~/4


For the non-bolometric case we have,
E (r) = K(r) (1-0 ) ff I(A,r,0)dXdw ,
a mA

giving us

1/4
T(r) = [(1/40) ff I(X,r,0)dXdw]


Because of spherical symmetry we can write this as

iT 1/4
T(r) = [(T/20) /f I(A,r,6)sin6 dX dO]
oA

If we divide the radiation field into two parts, as









we did before, we have

7T
T(r) = {(1/40)[2r f I (r,0)sin0 dO +
o

OT,4 T-TI 1/4
/ e sin6 do]} (22)
o


where Ic(r,O) is the bolometric intensity defined by equation

(15c). When the central star can be considered a point

source we have the additional equation,


T(r) = {(1/4a) [27 / I (r,O)sin8 dO +
o

(T-TI) 2 4 2 1/4
e (R*oT-/r2)]}


Note that in equation (9) I is a functional of S and

that in equation (10a) S is a function of j. We have seen

that j is a functional of I; therefore, I is a functional of

S and S is a functional of I. Thus, equations (9) and (10a)

can be rewritten as a recurrence formula for I(A,re,w), which

can be used in an iterative solution for the radiation field

throughout the cloud and hence for the desired spectral-

energy distribution.

The recurrence formula is


In(A,r,w) = fm et S[ I In_1(A,re,w)dw]dt, (23)
o o

where the subscripts n and n+l refer to iterations n and

n+l. The true radiation field consistent with a given set

of parameters will be given by


I(A,r,w) = lim T (,r,w)
n-t









This completes the definition and derivation of the

basic quantities needed to implement an iterative solution

of equation (11).

Once the proper set of the parameters, TI, R R and

n, is determined, we can obtain values for the density with-

in the cloud, p(r), and the total mass of the cloud M.

If we let D equal the number of particles per cm and

d equal the average diameter of a particle, where d < < R,

then


D = CSA/cm3 = particles/cm3
CSA/particle

The cross sectional area of the particles per cubic centi-

meter is a function of r and is given by K(r); thus,


D(r) = K(r) = 4 K(r)d-2
(2 2


For the volume and mass of a particle, Vp and m respective-

ly, we have


V= 4 d(d)3 cm3/particle,
p 3 2

and

m p ( ) grams/particle ,


where p is the specific gravity of a grain. The density of

the cloud at a distance r from the central star will then be

given by


p(r) = D(r) mp









4 d 3 K(r) or
p(r) n( ) p d or
3 2 P (d) 2



p(r) = p K(r) (d)


Substituting for K(r), we find

2 T (l-n)
p(r) p d [ ] 1 (24)
3 P Rl-n-Rl-n
I

From the above expression for p(r) we can derive an

expression for the total mass of the cloud. We have

R
M = I (47r2)p(r)dr ,
RI


from which we obtain,

M = 8ppTI(1-n) (R3-n R 3-n)d
3(3-n)(Rl l-(25)
3(3-n)(Rl-n Ri1-n)
















SECTION III
NUMERICAL PROCEDURE


In this section a procedure based on the equations of

Section II is developed, to effect a numericsl solution for

the radiation field in the dust shell. This procedure will

involve iterative numerical integration of equation (23)

to obtain a set of convergent series of n(A, ri,w) Here i

specifies the ith such set and ri, therefore specifies the

distance from the central star at which the I(A,ri,w)n for

that set are to be evaluated. The n signifies that this is

a value obtained in the nth iteration. All the integration

will be performed using Gauss-Legendre formulas.

In order to accomplish the necessary integration it

will first be necessary to establish an integration grid in

keeping with the boundary conditions imposed by the geometry

of the shell and the central star. Consider the spherical

shell of dust extending from RI to R and divided into N

thinner concentric shells each with the same optical thick-

ness, TI/N (see Figure 2 ). The inner and outer radii of

these shells are a function of T and can thus be found from

the equation


r(T) = [R1-n (T/T) (Rl-n r -n) 1(26)
l-n (26)









For each of the N points lying on the semi-diameter of

the cloud, SR, (see Figure 2) at distance ri from the central

star, we assign M values of the angle 6, which will define M

lines radiating from each point. We then break these

radial lines into segments of approximately equal optical

depth, where the end points of the segments are all the

points at which the radial lines intersect the N circles of

radii ri.

Thus, about each of the N points on SR we have M

radial lines, and each of these lines is broken into from 1

to 2N segments, where the actual number of segments for any

particular line depends on how many of the N circles of

radii ri that line intersects.

The M values of 8 used at each of the N points along

SR would in general be the Gauss-Legendre points for the

interval 0 to i; however, it is usually advantageous to

break this interval into two intervals with M/2 Gaussian

points each one from 0 to 9m(ri) and the other from 8m to ir,

where 6m(ri) is an angle that must be determined for each of

the N points ri. If ri is small enough so that the central

star shows an appreciable disk, then 6m(ri)=9*(ri), but if ri

is large enough so that the star can be considered a point

source Om(ri) will be an angle between 0 and T/2. The value

of 6m(ri) in this case should be determined, such that, it

lies between u/2 and the value of 9 at which I(ri,6) has its

greatest value. Defining 9m(ri) in this manner allows us to

concentrate our Gaussian points on the part of the interval









0 to a where I(0) is varying most rapidly. In one special

case the M Gaussian points must be determined in a different

manner. When ri = rn = R the M O's are restricted to the

interval 0 to f/2 since for i/2 < 6 < i, I(R,6) obviously

vanishes.

We have broken each of the M radial lines into segments

as a first step in the determination of the locations of L

Gaussian points to be used in the numerical evaluation of

equation (23). Figure (2) shows a few representative radial

lines for the ith of the N points on SR. Also, the radial

lines are shown broken into segments with end points a

distance sk from R, where sk is measured along the radial

line and k is an index ranging from 1 to 2N.

Let x and y be the rectangular coordinates of a point

in a right-handed coordinate system, where the central star

is the origin, and the semi-diameter SR is the positive

y-axis. Also, let the slope and y-intercept of a radial line

be denoted by m and b respectively; then, each pair of

values for m and b defines a particular radial line. In

terms of previously defined quantities, m = Cot (6) and

b = ri When b and m are specified, the rectangular coordi-

nates of all the points where a particular radial line inter-

sects the N circles of radii r. can be found by solving, for

x and y, the system of equations


x2 + y2 ri2 = 0


y mx b = 0, for i = 1 to N.









We are only interested in points to one side of SR, so we

exclude all solutions with x > 0. Points which are occulted

by the star are also excluded.

The starting point of a radial line will lie on SR,

and have coordinates x = 0 and y = ri; the kth point of

intersection on that line will have coordinates xk and yk'

Thus, the distance along the line from the starting point to

the kth point, which is denoted sk in Figure 2 is


sk = [xk2 + (ri-k) 2]-/2

To accomplish the integration of equation (23) we must

first perform the integration indicated by equation (12)

along each of the radial lines using the end points of the

segments on these lines as the limits of integration.

Equation (11) is used here to find the distance from the

central star to a point on the line and hence the values of

K(r) needed by (12) from (5) or (7a).

From (12) we will obtain for each of the N x M lines of

sight (LOS) a set of accurate optical depths, tk, and cor-

responding distances, sk, measured along the LOS. Because

of the discontinuity caused by the central region of the

dust cloud (r < RNR) it is helpful to break the LOS into two

intervals to consist of L/2 Gaussian points.

Unless the LOS intersects the central region of the

cloud (r = rl = RNR), the midpoint of each LOS, at which we

can break the LOS into two segments, is roughly determined

by dividing the total number of points by two. However, if









the shell rl is crossed two midpointss" are established

(say, MID1 and MID2) where the first such point is the point

of intersection of the LOS and rl closest to the starting

point and the second is the next point further along the LOS.

The segments from the beginning of the LOS to MID1 and from

MID2 to the intersection of the LOS with rl are to be

considered the two halves of the LOS (see Figure 2 ). One

additional case remains; if the LOS crosses the shell rl and

then intersects the surface of the star (r = R*) the LOS is

to be considered terminated at rI This will obviously

occur whenever 6(ri) < e*(ri) for a LOS.

Next each half of the LOS is broken into 19 segments

(this is the specific number used in the present version of

the computer code it can be varied if necessary) as nearly

the same length in optical depth as computing time will

allow (the computer code is given a parameter which sets the

precision desired). This is done by an iterative procedure

in which the computer code uses an Aitken-Lagrange inter-

polation routine (Hildebrand, 1956; Mullen, 1974) to inter-

polate in the table of sk and tk values for the set of sk

values corresponding to a new set of evenly spaced tk values.

A Gaussian integration routine is then used to evaluate

equation (12),thus redetermining the values of tk If the

values of tk found by this integration are not equally

spaced (to the precision requested) another interpolation

and integration are performed, and this process is continued

until the desired accuracy in spacing is achieved.









If the original set of sk and tk values contains only

two points (the case where the LOS starts at the outer

boundary rn and then intersects rn without intersecting

rnl) the procedure described above breaks down since a set

of two points can have no midpoint. In this special case

the code sets up a more extensive table containing 20 points

(midpoint at 10) by breaking the interval s2 to sl into 19

equal segments thus obtaining 20 sk values and then finds

the corresponding tk values by integration. The resulting

tables of 20 sk values (equally spaced) and tk values (un-

equally spaced unless n = 0 in density relation) are then

used in the iterative process to find the set of tk values

at equal spacing in optical depth.

Since we want to evaluate equation (23) by Guassian

quadrature we must be able to evaluate the integral at the

Gaussian points, tkg, on the interval 0 to 1,; therefore,

we next interpolate in our table of sk and tk values for

the sk values, designated skg corresponding to the tkg

values.

We have that S(t) = S(T(t)), where is the optical

depth measured from the outer boundary of the cloud to the

point specified by t; therefore, we want to find those

values of T which correspond to the tkg. From the skg

determined by interpolation we can calculate, using the law

of cosines, a set of corresponding rkg values and from them

a set of Tkg values. We can then find the values of S(Tkg)

by interpolating in a table of S(Ti), where Ti is the optical









depth at the ith of the N shells. All the interpolations

mentioned above are performed by the same Aitkin-Lagrange

interpolation routine.

The table of N values of S(Ti) is of course based on

the previous iteration of the code.

The computation to determine sets of T(ri), K(ri),

.S(ri,6,X), I(ri,6,X) and ultimately F(rN,X) starts from a

set of initial values of T(ri), which should be as close to

the expected final values as possible, and a set of the

presumed physical parameters. The set of K(ri) can be found

immediately from the given parameters using either equation

(5) or (7a). Using the K(ri), the initial set of T(ri) and

their associated optical depths, we can evaluate the integral

in equation (23) along each of the M lines radiating from

each of our N points. Then from the M values of I(ri,6,X)

obtained about each of the N points, improved sets of T and

S can be calculated.

This process is repeated until successive sets of

T(ri) converge to within a few tenths of a degree. The sets

of T, K, S, and I obtained in the last iteration are taken

as the true values for the cloud. A set of F(rN,A) is

obtained from the I(rN,6,X) using equation (8).

To save computation time a bolometric case with =

0.0 is run to determine the set of T(ri). If in the numer-

ical procedure, we are performing the calculations for L

values of A, then the computation time will be cut by a

factor of approximately 1/L when we drop the A dependence









and run a bolometric computation. When we use the bolometric

expressions instead of their wavelength dependent analogs,

equation (23) becomes the bolometric expression



0
I(r,9)n+1 = I e-t S[I(r,6)n] dt.



Like equation (23) this can be solved iteratively for the

specific intensity. The set of T(ri) is then found from

equation (22) and used as the set of initial values in the

desired wavelenth dependent solution.

Note that we can assume that the temperatures deter-

mined by a bolometric computation are the same as those

that would result from a detailed many wavelength computa-

tion only if the particles in the cloud are grey-bodies. If

we want to consider cases where wo = wo(X) then we can not

use a bolometric computation to determine the temperature

distribution.

The basic equations in Section II which involve inte-

gration over 8 can be rewritten such that the integration

are over p = cosO by making the change of variable

a = cos-l (); dB = dp/sin6 The code has been tested

with these equations programed with 6 as the variable of

integration and with cos6 as the variable of integration -

the results do not differ significantly.

When the central star can not be considered a point

source it is necessary to integrate over the angular diameter

of the star. Since the specific intensity of the stars'sur-

face can be considered constant (we can ignore the small









effects of limb darkening) an integration from 6 = 0 to

8 = ,*(ri) need only be done once for each ri and used in

place of the relevant point source approximation.

For a shell very close to the star the correct contri-

bution by the star to the mean intensity and the approximate

value can differ by as much as a factor of 2.0 the point -

source approximation giving the lower value. The mean

intensity, J, the Eddington flux, H, and the so-called K-

integral, K, are all moments of the mean intensity I (Mihalas,

1970) that is, M(n) = 1 I()PndVJ,
2-1
where I = M(0), H = M(1), and K = M(2). The flux F should

not be confused with H; they are related such that F = 4wH.

When RNR >> R* we need not worry about inaccuracies due to

approximating the star as a point source. For example, when

R, % 0.2 x 1013 and RNR 4 0.15 x 1014 we find an error due

to the approximation of only 0.46%.

Since the code is capable of evaluating the specific

intensity as a function of 6 and X at any point in the shell

it can be used to find values of Jx(T), H (T), Kx(T) and the

Eddington factor, KX(T)/JX(T) The Eddington factor can

vary from 0.0 to 1.0 but is usually assumed to be 1/3 (the

Eddington approximation, K = 1/3 J) since this is the value

it would have in an isotropic radiation field.















SECTION IV
RESULTS OF MODEL CALCULATIONS


The computer code used to determine the radiation

field within a circumstellar dust shell is broken into

three main programs (program 1, program 2, and program 3).

The first is a program used to set up the integration grid

and store it in the computer (usually on disk storage), the

second is a program used to determine the bolometric radia-

tion field within a dust shell and to find the temperature

distribution, and the third program is used to calculate

the monochromatic radiation fields for a number of wave-

lengths using the temperature distribution determined by

the second program. The three programs must initially be

run in the order given but program 1 need only be run once

for each set of the geometric parameters R., RNR, R, TI,

and N. Therefore it is possible to run a number of differ-

ent models with the same integration grid by varying the

two nongeometrical parameters T, and o .

The models that we want to consider here are charac-

terized by the parameters in Table 1.










Table 1. Values of the seven parameters used for each of
the models computed (cgs units; T* in oK)



Model R, RNR R TI n Wo T,


I .202x1013 .15x1014 .75x1015 5.0 1.5 0.1 6000

II 0.0 "

III 1.0 "

IV 0.6 "

V 7.0 0.0 "

VI 0.1

VII .202x1013 .202202x1013 0.0

VIII .625x1015 .75x1015 2.0 2.0 0.0

IX .15x1014 5.0 1.3 0.1

X .8352x1012 .8352x1015 3.97 0.0 0.8 12500




One additional model, model XI, will be described later in

this section.

From Table 1 we can see that the most common values of

Wo used are 0.0 and 0.1. This is because the amount of

computing time necessary to determine the radiation field

is a function of o For small values of o()o<.2) program

3 requires approximately six iterations to determine values

of the flux, for example, converged to three significant

figures, whereas, if Mo = 1 the program will need at least

15 iterations to achieve the same accuracy. Because of the

high cost of models with large values of wo only two are










computed, models IV and X. Model III was computed for only

one wavelength and therefore is not comparable in cost to

models IV and X which were computed for 11 wavelength. When

to = 1.0 we can save time by running the program for only

one wavelength, Al say, and then multiplying the quantities

obtained for that wavelength by the factor B(12,T,)/B(AI,T*)

to obtain the quantities appropriate to X2. This is a

valid procedure when wo = 1.0 since under that condition all

the radiation leaving the star must reach the surface of the

shell unchanged in its spectral-energy distribution.

When program 2 is run the total outgoing bolometric

flux through each shell is computed at each iteration. Since

the only source of radiation is the central star, the total

outgoing bolometric flux at each shell should be equal to

the bolometric luminosity of the star. This is true to within

0.5% for some of the models in Table 1 and to within 4% for

all the models in Table 1 except model VII. The flux

(bolometric) emitted by the outer boundary of the shell in

model VII is too low by 23% principally because of the very

severe geometry used in model VII. This model was run as a

test of the code to determine its accuracy under extreme

conditions.

To determine the temperature distribution in the shell

program 2 is run. The temperature distributions for the

models in Table 1 are shown in Figure 3. Note that the

curve for model IX is very nearly a straight line. This is

due to the geometry of the shell and the fact that, as we










shall see, the radiation field in the shell is highly

anisotropic. For a highly anisotropic radiation field all

moments of the intensity are equal, that is J = H = K. Thus,
1/4 1/4
J F and we know that T a J/ so T F/4. Because of

conservation of flux in a spherical shell we have that

F l/r2; therefore, T (1/r2)1/4 or T l/r05. There-

fore, for n = 1.5 we have from equation (6) that T(r) =

C1 + C2/r05, where C1 and C2 are constants. This means

that

T(l/r0.5) = C1 + C2 (l/r05)

and, therefore we can see that T will be a linear relation

of T.

The temperatures of the dust grains as mentioned in

Section I are consistent with the assumption that they are

carbon grains where the temperature is below about 21000K

and consistent with the assumption that they are refractory

silicates for temperatures less than about 1500K. Note

that the spread in temperatures at T = 0 is small even

though some of the models have very different physical

parameters.

Program 3 is run to determine a variety of quantities

for the cloud. These quantities are: I(X,T,6), J(A,T),

K(X,T)/J(X,T), and F(A,T). Figures 4 and 5 show some

representative values of these quantities (computed for

model IX). The conversion of visual to infrared radiation

can be seen on Figure 4 by following the relative intensi-

ties of a few wavelengths from T = 7 to T = 0. Figure 5









also shows this by way of the graphs for J and F. The graphs

for K/J show that the radiation field is generally anisotropic

although at depth the infrared radiation field does approach

the Eddengton approximation, K/J = 1/3. Thus, the use by

Huang and Apruzese of the Eddington approximation can be ex-

pected to decrease the accuracy of their results significant-

ly.

In Figure 6 the run of temperature with optical depth

is shown for two models calculated with this code and for two

models calculated by Huang (1971). The two models calculated

with this code are models VIII and XI. Model XI has the fol-

lowing parameters: I = 2.32, R* = .202 x 1013 cm,

RNR = .609 x 1015 cm, R = .75 x 1015 cm, n = 2.0 and

T* = 60000K. For optical depths larger than about 2/3 the

curves for models VIII and XI are very similar the curves for

Huang's corresponding models. From optical depth 2/3 to 0

the curves for models VIII and XI drop off much more rapidly

than Huang's curves. This is just what we would expect to

happen due to the erroneous value of the Eddington factor

used by Huang for the outer part of the cloud.

Figure 7 shows the spectral-energy distribution for

HD 45677 as given by Swings and Allen (1971) and Low et al.

(1970). On the same graph the spectral-energy distribution

for model X is also plotted. Model X is based on a set of

parameters derived by Apruzese (1974) for HD 45677. He

determined these parameters by fitting the observations in

the visual since his model is not capable of directly









fitting the infrared observations (actually his model does

use the infrared data in that he attempts to make the total

infrared emission calculated by his model equal to the

total infrared emission given by the observations). The

graphs show that his parameters do give a very good fit in

the visual region and that the total emission in the

infrared predicted by his parameters is approximately cor-

rect; however, his parameters do not give a good fit to the

actual infrared spectral-energy distribution. A different

set of parameters could be found that would fit both the

visual and the infrared regions.

Figures 8 and 9 show the spectral energy distributions

for a number of the models listed in Table 1. On Figure 8

the graphs for models II, V, VIII and X are shown. The

curve for model VI would fall between those for models II

and V. We can see the effect of changing the various model

parameters by studying these curves.

The curves for models II and V show that increasing

the total optical depth of the shell causes the peak of the

distribution to move toward the red (Figure 3 shows that at

the same time the temperature of the inner part of the shell

goes up) and causes the amount of visual radiation filtering

through to decrease by a large amount.

Models VIII and X show, as we would expect, that when

the optical depth of the shell is decreased by a large

amount the central star causes the distribution to peak in

the visual.










Figure 9 shows the spectral-energy distributions for

models I, IV, IX and the means of the observations of R Mon

due to Mendoza (1966, 1968), Low and Smith (1966) and Low

et al. (1970), where vertical bars show the amount of

variability from those means. The vertical error bars on

the observations at 20 and 22V indicate the root-mean-

square error of a single observation, approximately 20%.

The U through M values plotted are intensities

calculated from Mendoza's observations (corrected for inter-

stellar extinction), using Johnson's (1966) calibration of

the UBVRIJKLM system. The interstellar extinction correc-

tion was made, assuming that R Monocerotis is located in

NGC 2264, and using the following adopted data for that

cluster and hence R Monocerotis: EBV = 0.06 (Strom et al.

1971) and a reddening curve of the NGC 2244 type with R = 5.4

(Johnson, 1968).

Using these values we have AV = 0.324, and from the

reddening curve we obtain the extinction in magnitudes, AM,

for each of the observed colors (see Table 2).

The root-mean-square errors of a single observation

are shown for the 201 and the 22p observations in Figure 9,

but for all the other bands the root-mean-square errors of

a single observation are negligible, considering the

intrinsic variability of the object. The largest such er-

ror is for the M band, and it amounts to less than 0.1

magnitudes.









Table 2. Interstellar extinction in magnitudes, Am, for R
Monocerotis in each of the colors observed



Band Am


0.424

0.383

0.324

0.277

0.224

0.189

0.141

0.094

0.083


The curve for model IV shows that a large increase

in the value of wo causes the peak of the distribution to

narrow and move toward the blue, it also shows that the

direct contribution of the star in the visual increases

greatly. The curves for models I and IX show that a small

increase in the index n causes the peak of the distribu-

tion to move toward the blue, while the ends of the curve

in the visual and far infrared drop.

Model I is the best fit to the observations of R Mon

but it looks as though Model IX with a slightly larger wo

would fit somewhat better.

The following discussion pertains to the development

of the model for R Mon.









The spectral class of R Monocerotis, roughly G (Joy,

1945), and luminosity of 870 L (Low and Smith, 1966) lead

to the values of T* and R, chosen for these values since
4
4IRR*oT* 870L The value for TI was determined from a

cursory examination of the observations in the following

manner. We have the ratio of the visual flux, FV, to the

bolometric flux, F for R Monocerotis; namely, Fv/FB -.006

(Low and Smith, 1966). From this and the relation

TI = Zn(FV/FB) we obtain for T(RI); TI ~ 5.

It should be noted that this method gives only a

lower limit to TI, since we cannot say what portion of FV

is actually coherently scattered radiation. Thus, wo and

T cannot be determined separately with any ease. The

values of w0, RI, R and n used were just rough initial

approximations.

We must consider the following observational data

for R Monocerotis: (a) the spectrum does not show selective

absorption (Joy, 1945), (b) strong variable optical

polarization exists indicating scattering by large grains

(Zellner, 1970), and (c) irregular light variations occur

of several magnitudes (Joy, 1945; Mendoza, 1966, 1968; Low

et al. 1970). In addition we have the UBVRIJKLMQ observa-

tions of Mendoza, Low and Smith, and Low et al. mentioned

previously.

Points (a) and (b) above are consistent with the as-

sumption that the cloud is made up of solid particles which

act as grey absorbers and emitters. For a period of










variation of the polarization which is fairly short, the as-

sumption that the particles are randomly oriented should be

reasonable. Because of the variability mentioned in point

(c), those observations in several colors, which were made

on the same day are of the greatest interest, and where such

observations are lacking, we are forced to compare the

computed fluxes to mean observed fluxes.

Using the parameters from Case I given by

TI = 5 ,

RI = .15 x 1014 cm

R = .75 x 1015 cm,

n =1.5,

and assuming the particles to have the density of carbon or

silicon dioxide; that is, pp ~ 2.0, we have from equation

(25) for the mass of the cloud.

M = 13dM


This gives us for particles about twenty microns in

diameter a cloud mass of about 10-2 M .

The set of parameters for Case I also gives us for

equation (24) the following form:
d
p(r) = (1.5 x 109) r1.5


If d = 20p and r ranges from 1013 to 1015 cm, then p(r)

will range from approximately 10-13 to 10-16 grams/cm.

In summary, the best fit to the observations was

obtained by a model with the following parameters:

RI = .15 x 1014 cm (-lAu) ,


__







66



R = .75 x 1015 cm (-50 Au)

T =5,

= 0.1

n = 1.5

M~ 10-2M


10-16 < p(r) < 10-13 gm/cm3


2250K < T < 2,1000K .


~
























Radiation is shown falling on point 1 at
distance r from the central star. It is
incident there at an angle 01 after being
emitted from point 2 at an angle 0 .Part
of this emitted radiation is radiation
from point 3 which was incident on point
2 at an angle e2, and scattered through
an angle a. Point 2 is a distance re
from the central star and a distance s(t)
from point 1. Analogous triangles, with
vertices at the star, point 1 and any
point on the line directed from point 1,
at the angle 81, can be constructed.

Note that the angle 61 is designated
simply 6 in the text.


Figure 1.























/
/
/
/

/
/

/
/
/


-
f- -


STAR





























A schematic diagram of the integration grid
for the case when N = 4. Four lines of sight
are shown to indicate the manner in which
the boundary conditions at rl and rn are
handled. The filled circles on the lines
represent the initial grid of intersection
points. The tick marks on the lines
represent the points found at equal inter-
vals of T by an iterative technique. The
open circles represent the Gauss-Legendre
points for a three point quadrature formula.


Figure 2.



















U)







0
-H






0)






i-,-
a)











0-
'iU
> El






t-I
.10





>


F 0
Qo









.rl
CO
H a




0-o
- m





OCi
ta







-I






re
00
H-i
Os-
.5-La












n-














































0
0
H-


























Log [I(x,6,T)] vs. 6 for model IX. The angle
0 is in radians. The three panels from top
to bottom are for T = 7.0; 3.68; 0.0. The
curve are labeled with letters designating
their respective wavelengths on the
UBVRIJKLMNQ system. Note that in the case
for T = 7.0 the star is not considered a
point.


Figure 4.














v

M

r-- 7.0
13


12 K M






10






11 : 3.6 8



1M M


10 .










t 2



























Figure 5. Log [J(A,T)], K(X,T)/J(X,r), and Log [F(X,T)]
vs. T for model IX. The curves are labeled
with letters designating their respective wave-
lengths on the UBVRIJKLMNQ system.



















km
- a)HOa)H r
rd x ? a) o a c a)
0 O4O a 0a4- n
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a 0 00 OWOOd
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E-1 4 ) 0 ,C 4 U044
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H 0 0n 0 *H 0 H 0a N
H -C0- 4 a r 0 H OO
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O4 O P 0 H 00 4- OH
O) O H -H w r: H
4J *a oa x l OnN (' OHr
10 a rl Icl 0-H (- d H Win a, ^
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aN W )k a (d W E 140
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4a I ) a) (1) p >'44 r-ic
rWWEn Z0 > 04 00 0
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N P d ) 4-N -0H -4 U0 > -r0
r-H H (Z 12O44 l0) 44 a lJ
N 400 (0 O :--I- Q4-
(d U aWr) PE 4 H 3r--H a OO
ER--l 124 a) d > P 0U Q 4-3
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0 04 0 rd 9 Z nd ni 0 0 X 0
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LIST OF REFERENCES


Ackermann, G., Fugmann, F., Hermann, W., and Voelcker, K.
1968, Z. Ap., 69, 130.

1970, Astron, Ap., 8, 315.

Anderson, L., and Kuhi, L.V. 1969, Non-Periodic Phenomena
in Variable Stars, ed. L. Detre (Dordrecht-Holland:
D. Reidel Publishing Co.), p. 93.

Apruzese, J.P. 1974, Ap. J., 188, 539.

Breger, M. 1972, Ap. J., 171, 539.

Breger, M., and Dyck, H.M. 1972, Ap. J., 175, 127.

Capps, R.W., and Dyck, H.M. 1972, Ap. J., 175, 693.

Chandrasekhar, S. 1934, M.N.R.A.S., 94, 444.

Chavira, E. 1967, Bol. Obs. Tonantzintla y Tacubaya, 4, 29.

Cohen, M. 1973a, M.N.R.A.S., 161, 85.

1973b, ibid., p. 97.

1973c, ibid., p. 105.

Gilman, R.C. 1969, Ap. J., 155, L185.

Grasdalen, G.L., and Gaustad, J.E. 1971, A.J., 76, 231.

Hayashi, C. 1966, Ann. Rev. Astr. and Ap., 4, 171.

Herbig, G.H. 1960.

1968, Ap.J., 152, 439.

1970, ibid, 162, 557.

Hetzler, C. 1937, ibid., 86, 509.

Hildebrand, F.B. 1956, Introduction to Numerical Analysis
(New York/Toronto/London: McGraw-Hill), p. 49.

Huang, Su-Shu 1969a, Ap. J., 157, 835.

1969b, ibid., 157, 843.









Huang, Su-Shu, 1971, ibid., 164, 91.

Hummer, D.G., and Rybicki, G.B. 1971, M.N.R.A.S., 152, 1.

Hyland, A.R., Becklin, E.E., Neugebauer, G., and Wallersten,
G. 1969, Ap. J., 158, 619.

Johnson, H.L. 1966, Ann. Rev. Astr. and Ap., 4, 193.

1968, Nebulae and Interstellar Matter, ed. B.M.
Middlehurst and L.H. Aller (Chicago: University of
Chicago Press), p. 202.

Joy, A.H. 1945, Ap. J., 102, 168.

Krishna Swamy, K.S. 1970, Ap. and Space Science, 9, 123.

Kukarkin, B.V., Kholopov, P.N., Efremov, Yu, N., Kukarkina,
N.P., Kurochkin, N.E. Medvedea, G.I., Perova, N.B.,
Federovich, M.S., and Frolov, M.S. 1969, General
Catalogue of Variable Stars, Third Edition (Moscow).

Larson, R.B. 1969a, M.N.R.A.S., 145, 271.

1969b, ibid., 145, 297.

Low, F.J., and Smith, B.J. 1966, Nature, 212, 675.

1970, Semi-Ann. Tech. Rep. Contract No. F 19628-
70-C-0046 Project No. 5130 (ARPA).

Low, F.J., Johnson, H.L., Kleinmann, D.E., Latham, A.S.,
and Geisel, S.L. 1970, Ap. J., 160, 531.

Mendoza, V.E.E. 1966, Ap. J., 143, 1010.

1968, Ap. J., 151, 977.

Mihalas, D. 1970, Stellar Atmospheres (San Francisco: W.H.
Freeman and Company), p. 5.

Mullen, J. 1974 (Private Communication).

Neugebauer, G., Becklin, E., and Hyland, A.R. 1971, Ann.
Rev. Astr. and Ap., 9, 67.

Neugebauer, G., and Leighton, R.B. 1969, Two-Micron Sky
Survey a Preliminary Catalog (NASA SP-3047).

Poveda, A. 1965a, Bol. Obs. Tonantzintla y Tacubaya, 4, 15.

1965b, ibid., 26, 15.

1965c, ibid., 26, 22.










Prentice, A.F.R., and terHaar, D. 1971, M.N.R.A.S., 151, 177.

Serkowski, K. 1971, Proceedings of the Conference on Late-
Type Stars, ed. G.W. Lockwood and H.M. Dyck (Kitt Peak
Contr. No. 554), p. 107.

Stein, W. 1966, Ap. J., 145, 101.

Strom, K.M., Strom, S.E., and Yost, J. 1971, Ap. J., 165,
479.

Strom, S.E. 1972, PASP, 84, 745.

Strom, S.E., Strom, K.M., Brooke, A.L., Breger, J., and Yost,
J. 1972, Ap. J., 171, 267.

Swings, J.P., and Allen, D.A. 1971, Ap. J., 167, L41.

Walker, M.F. 1969, Non-Periodic Phenomena in Variable Stars,
ed. L. Detre (Dordrecht-Holland: Reidel Publishing Co.),
p. 103.

Walker, M.F. 1972, Ap. J., 175, 89.

Woolf, N.J., Stein, W.A., and Strittmatter, P.A. 1970,
Astron. Ap., 9, 252.

Zellner, B. 1970, A.J., 75, 182.

Zellner, B.H., and Serkowski, K. 1972, PASP, 84, 619.















BIOGRAPHICAL SKETCH


Christopher Alvin Harvel was born on December 7, 1944

in Columbia, South Carolina. He attended public schools in

North Carolina, California, New Jersey, and Virginia, gradu-

ating from McLean High School in McLean Virginia in 1963.

Four years later, in 1967, he received his Bachelor of

Science degree with a major in Astronomy from Georgetown

University in Washington, D.C. and started graduate school

at the University of South Florida.

In December of 1968 he was drafted into the United

States Army and in December of 1969 he began a one-year tour

of duty in the Republic of South Vietnam.

At the end of his military service in December of

1970, he returned to the University of South Florida and

received a Master of Arts degree in Astronomy in 1972. In

that same year he started graduate school at the University

of Florida where receipt of the Ph.D. degree should occur in

August, 1974.










I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.






Frank Bradshaw Wood, Chairman
Professor of Physics and
Astronomy






I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.






Sabatino S. Sofia
Professor of Astronomy
University of South Florida






I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.






K hen
Kwah-Yu en
Associate Professor of Physics
and Astronomy










I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.






Howard L. Cohen
Associate Professor of Physical
Sciences and Astronomy






I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.






James W. Dufty 1-P
Assistant Professor of Pnysics









This dissertation was submitted to the Graduate Faculty of
the Department of Physics and Astronomy in the College of
Arts and Sciences and to the Graduate Council, and was
accepted as partial fulfillment of the requirements for the
degree of Doctor of Philosophy.

August, 1974





Dean, Graduate School




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