Bull Astr Soc l n d ~ a (1990) 18, 119-135

Abhyankar-The scientist

A. Peraiah Indlan Inst~rure of Asrroph.~sics. Bangabre 560 034

1. Introduction

There are few people who can study both observational and theoretical astronomy. Few of them are successful and still fewer are successful in the face of adversity. Krishna Damodar Abhyankar belongs to 'such a selective group.

He is passionately devoted to astronomy He has taught and spread astronomy with a missionary zeal. His lucid exposition of intricate ideas makes his lectures so interesting that even a nonexpert in the field enjoys them immensely. He has got a thorough understanding and deep knowledge of astronomical obsenatlons and a ~c,rrc\p~,ni l~ng keen sense of theoretical lasight to unravel the secrets shrouded in the observations which look like mere numbers.

He is not only a successful teacher but also an Indefatigable research enthusiast. He has taught many students at graduate, post-graduate ievels a t the Department of Astronomy, Osmania University and guided several students for M. Phil and Ph.D. degrees. He was solely responsible for introducing astronomy at the graduate and post- graduate levels, arranging the experiments, writing the syllabi of various courses, etc. I have had the fortune of being associated with him a t the very early stages of my research career and observed him from close quarters. 1 benefitted immensely from this association with him and learnt how meticulous one has to be in doing research. 'There is now what is called the Abhyankar school of students, many of whom are working a t different places in the country for instance at the Indian Institute of Astrophysics.

He is singularly responsible for the establishment of the Astronomical Society of India. He served it in several capacities iqcluding its Presidentship. This organization is playing a major role in bringing ideas together and coordinating programs, which is a great service to astronomy in the country.

He was born on 1928 June 21 -the day of the summer solstice-an astronomically significant day. Since his childhood, he showed signs of brilliance. He secured gold medals a t high school and colleges. After he finished his M.Sc. from Agra University, he worked for a short period a t Holkar College, Indore. He secured a senior research fellowship of the government Bf India and joined Kodaikanal Observatory during 1952- 1954. He worked there with the then director Dr A. K. Das to work on the problems of the sun. He secured a scholarship at the University of Czlifornia a t Berkeley for studying towards his Ph.D. degree. He was asked to take 16 courses in astronomy and he obtained A grades in all these courses-a record which stands unbroken till today! Here, for his thesis, he studied several close binary stars with a view to understanding stellar evolution. In the course of his studies, his thesis supervisor was Otto Struve, a giant among the 20th

c:z~:~!:! ,!~,::irr\~,ln:is lie d15corr.red 3 short per~od l a r ~ a b l r AD CMi HIS I'h D. the5is 160rh I ~ i r ~ d 021 lo be a n~a,t.c.rplrce and it has been \cry well quoted.

tir. icturn;.d to Ir,dla brlmmlng with enthuslasrn to spread and pursue research In listror.oy:~> H e jo~ned as meteorolog~st grade 1. a t the Kodaikanal Observatory from i\hc:c 11c Icit for t lS 4 icu >ears car her 'Th15 posrt~on could neither sat~sfy h ~ s des~rt: for re\rarch nor contam his enthuslahrn to teach astronoml \Vhen there was a pos l t~on

\acdnt jn thc nett I! started department of astronomy at Osrnanla Umverslty, Hyderabad, he ~mmcdlnr~l! pried this department. He xas one of the few astronomers at that tlme who kncl.+ u hat should be done In the initial stages of startlng an astronomy department. He trorhco crasele~al> n ~ t h unbounded enthusiasm, foresight and thought in p repa r~ng the set of course work. obszrvat~onal programs for the graduate and post-graduate levels, ::~:.::.,:1; people for \iirious posltlona, and d o ~ n g administrative work. In a d d ~ t ~ o n to these dut1c.s he conducted site survey for the proposed 48" reflector and selected the present place near Rangapur xlllage.

When the former dlrector of the Vizarniah Obsenratory Dr Akbar Ali died. Dr A. K. Das the former director of Kodaikanal Observatory, who had retired not long ago, was temporaril) appo~ntcd the director He too was snatched away by the icy hands of death \hithin a short time Abhyankar was made Incharge director of the N ~ z a m ~ a h Obserkatory and head of the a\tronomy department. D u r ~ n g t h ~ s short period of his in- charge Directorship, he brought up the department to the level of a standard that can be compared with standards anluhere in the world He managed the department with a skeleton staff and he hlmself used to take several classes for B.Sc. and M.Sc. during the day time, and conduct observations during the night t ~ m e . It was a place b r ~ m m i n g with enthusiasm and ekeryone in the department and observatory was enthralled to work with h ~ m . Suddenly, the control and d~rection of the observatory and the department was snatched a u a y from h ~ m and passed into hands which showed only darkness. I t was like passing from brilliance into darkness. One can notlce that with in a short period the change that took place all that he had so ass~diously built up became redundant and a n object of ridicule. Astronomy suffered a set back at Osmania.

He left for Canada to work at the David Dunlap Observatory on NRC post-doctoral Fellowsh~p for a year in 1963 He started work on the problems of moving atmospheres. He continued to work on the problems of binary stars. He was made a full professor upon his return from Canada. However, he was unable to pursue research under suffocated circumstances. He proceeded to USA in 1967 on a senior post-doctoral resident research associateship of NRC-NASA of USA a t Je t Propulsion Laboratory, Pasadena. I n collaboration with A. L. Fymat, he wrote a series of papers on the solutions of the radiatike transfer equation. They solved this equa t~on by perturbation techniques In inhomogeneous and non-conservative scattering media. After his return to India, he took a student R. K. Bha t~a and they extended this technique to the study of planetary atmospheres.

He served as the director of Japal-Rangapur Observatory and as head of the astronomy department during 1961-63 (incharge) 1973-81, 1986-88, till he retired in 1988. He served the faculty of science of Osmania University in the capacity of dean during 1977-80.

He has been elected Fellow of Royal Astronomical Society (1964), lndian Academy of Sciences, Indian National Science Academy, Member of Sigma Xi of U S A in 1955,

Member of Astronom~cal Soc~ety of the Paclflc. Xmcr~can Astr onnmrcai Soc~et! . International Astronomical Un~un He 1s llle Foundcr m c m b e ~ o l -L t ronon l~ca l Socletb of India, Andhra Pradesh Plcudem) of Sciences, and Llaharahtra Acadern! of Sclcnscs.

He had been akarded the best tcacher anard by the Xndhra Pradesh Go\ernment. and NSSA ahard for Patent R~ghts on a nsn techn~qus of rncttiur~ng opt:ca; polarization.

2. Scientific achievements

His scientific work can be divided into 3 main areas of activ~ty: (1) solar physics, (2) binary stars and variable stars, and (3) radiative transfer and planetary atmospheres.

It is practically impossible to narrate the whole work here and instead we shall highlight a few points from each of these areas

1. Solar phjsirs

Us~ng Bjerkness thermodj'namical theory of sunspots, the depth of the sunspot umbral column is calculated from the observed d~fferences In the temperatures at the pole and equator. Bjerkness relatlon 1s

where t9 is the surface temperature of the sun; V the circumferential velocity at equator equal to 2000 ms-'; VE the eastward velocity of the highest circulating stratum, g the acceleration due to gravity; A6 the difference in temperatures at pole and equator of the sun; and h the thickness of the upper circulating stratum above subphotospheric zonal vortex in the lower stratum in which the sunspot originates.

In collaboration with A. K. Das at Kodaikanal, Abhyankar estimated the temperature difference between pole and equator of the sun by measuring the equivalent widths of the lines A 4227 A of neutral Ca and A 3933 A line due to ionized Ca atom at the pole and equator [ I , 2]*. They employed W oolley's method and found that the- pole is hotter than the equator by about 96K k 18K by employing Pamekeock's continuum absorption coefficients, and the difference is 86 + 16K with Chandrasekhar's values. Using Bjerkness theory, they derived 125 km or 140 km for the depth of the sunspot umbral column, using the above observed values of temperature differences.

In collaboration with A. S. Ramanathan [3] he determined the excitation temperatures of 4030K, 4200K and 3800K for Cr I, Fe I, Ti I respectively from the spectra of sunspots by using curve of growth method.

In collaboration with K. Anthony Raju, P. V. Subramanyam, he studied several aspects of the solar corona at the total eclipse of 1980 February 16.

*?he numben m brackets refer to those in the list of references.

122 A. Peraiah

2. Binary stars, variable stars and rebred probkms

He has studied the stability of straight-line solutions in the restricted three body problem and concluded the falioning [9]:

(i) Over a wide range of initial distance from Lz, even with the proper initial conditions, the particles make only one revolution around L2 before they go off. Particles moving near L3 make two rounds if the distance from L3 is sufficiently small. The orbits are partly elliptical and their periods and eccentricities are in fair agreement with the predictions of the first-order theory.

(ii) The incipient stability is destroyed even if the initial velocity components differ by about 1% of the velocity components. For bigger deviations the orbits depart more and more quickly from the elliptical orbits of the first-order theory.

(iii) These results are not favourable for accumulation of material near the Lagrangian points Lz and LP. Gould had found that the particles leaving either component of a binary system stay near Lz and L3 only for a short time during which their orbits am changed from direct to retrograde motion.

(iv) Most of the computed orbits are funnelled into orbits close to each other, which surround both components of the binary system. This result gives support to Gould's conclusions regarding the formation of rings in close binary systems.

(v) A few orbits were computed for the case p = 0.5. The results were similar to those of L2 in the case of p = 0.1.

He worked for his PLD degree in the University of California at Berkeley under the guidance of Otto Struve. He decided to study binary stars because of the fact that these are believed to give clues to stellar evolution. In the course of his Ph.D. work, Abhyankar wanted to determine the light curve and orbital elements of AD CMi whose light variability was discovered by Hoftmeister and confirmed later by Zessewitch. Abhyankar noticed that the observations did not indicateathat this is a binary but an ultra short period variable with 3 hr period. Further observations [ I l l by him conclusively proved that AD CMi is a variable (figure 1).

In h i s ' p h . ~ thesis [8] he studied three binaxy systems. The purpose of this study was to obtain more information at different stages of stellar evolution by examining their internal structures. For this purpose he chose systems f l Sco,.AO Cas and HD 47129 ( ~ b k e t t sm) (see figures 2, 3 and 4 respectively for their radial velocity curves). He obtained the following results for the three systems.

The internal structure in terms of density distribution is determined by the motion of the line of apsides. From the observations he cakulated the periods of the line of apsides and rtvolutionary period of the system. The relation by T.E. Sterne connects the inkmal structure of the component stars with the apsidal motion through the equation

where P and U an the periods of binary revolution and the apsidal line respectively; m ~ , r n ~ the masses of the components; R, andRt the radii of the components; r is the distance bttwten the stars; and KI and KZ are the apsidal constants.

Abhj,ankor- The sclentrsr

Figure 1. The light-curves of AD CMi. The small dots represent observations on four nights: January 30, February 2. February 3, and February 5, 1959. The larger dots represent observations on three rughts: February 28, March I , and March 2. 1959. The mean epochs for the two sets of observat~ons are separated by about 220 cycles.

Assuming that the two components are similar in their evolutionary characteristics, he set KI = K2 = K. By employing the equation of T. E. Sterne, he derived a value of 2.82 X for the value of K. This corresponds to n, the polytropic index between 3 and 4. The central condensation defined as p,/ p (where p, and p are the central and average density of the star) was found out to be between 50 and 600.

Figure 2. The rdd:dl \eluclt!: cunes tor P Sco Open circles (pr~mary) and crosses (secondary) represent measures on more recent \fount U'llson plates.

Figure 3a. The radial belocity curves for A 0 Casslope~a~.

- 3200 -2400 - 1600 a00 0 €

Figure 3b. Aps~dal mot~on In A 0 Casslopelae

Figure 4. The radlal velocity curves for HD 47129

, 4 0 Cas

The bame procedure as In the case of /3 Sco is applied ro this system for obtaining p,]p and the apsidai constants have been derived to be Ki = Kz= K = 0.00065 and n - 4 which gave the central condensation to be about 600.

HD 41129 (Plasketr star)

In the case of Plaskett star he denved the mean d~stance of the envelope and other factors were calculated. Radius of the envelope = 6Ro. The abundances of H, He, n, are found to be n~ = 1 3 X 10" cm-', nH, = 0.2 X 10" cm-), n, = 1.7 X 10" cm-'.

He studied the spectroscopic binary f CrB A, incollaboration with M. B. K. Sarma [22]. They showed that ml jmz is less than one; and that the secondary has a variable velocity curve and variable line intensities. These observations show that this system might now be in the earliest phase uf its development into a stage where the secondary component becomes more and more unstable.

He obtained elements of several binary systems with UBV observations such as YY LCMi [13, 141, RV DCorvi [48], WX Wridani [55, 581, TT Hydra [57, 58, 62, 701, TT Aurigae [76], RC Ma [8 1 , 90, 911 in collaborstion with his colleagues M. Parthasarathy, 3. B. Sanwal, M. B. K. Sarma, A. G. Kulkarni, J. Kaul, N. Rajasekhara Rao, £3. Lakanandhan, K. R. Radhakrishnan. He studied the period changes in the eclipsing binary stars [63, 65, 68,71, 75, 84,87,92,93] in collaboration with T. Panchatsaram, T. R. Radhakrishnan, M. B. K. Sarmil, R. K. Bhatia, P. Devadas Rao and C. V. S. Sanna. He and Praveen Nagar proved that the secondary components of the Algol systems are more luminous and hotter for their mass, indicating their hydrogendeficient character [82, 943.

3 . Radiative trans/er

In the early sixties the art of simulating absorption line profiles in moving envelopes of outer layers of stars was at a primitive stage. Abhyankar extended the- work of S. Chandrasekhar and developed a numerical method for computing theoretical absorption line profiles in a moving atmosphere [19, 20, 211 during 1963-64. This is a simple and easy to understand techhique in which the medium is divided into N discrete layers (see figure 5). He computed a series of absorption line profiles for different scattering functions and optical depths (see figure 6). Eater he performed the disc integration of the profiles which makes the profile smooth 1241 (see figure 7).

Abhyankars' work [32, 331 with A. L. F y m t has already been referred to. If it is assumed that the albedo for single scattering a(r) differs from a ionstant

value no by a small amount

throughout the atmosphere, the nonlinear singular integral equations for the X and Y functions of Chandrasekhar, and the X and Y functions of Ueno, which describe the transfer of radiation through and inhomogeneous pkne-parallel atmosphere of arbitrary stratifition, are linearized by use of a perturbation technique. These equations are written in their operator form, and their itemted solutions are expressed as infinite series

Abhyankar- The screnasr

Top of the otnrn~phere ?'v= '4 N \ Z'Z,, T'T, X = X ,

\ J

I / \ /FI N(0.v)

Photosphere: ZED, T - 0 , X r O FoUt[l,U]= I\

Figure 5. Divis~ons of the atmosphere Into K layer, and defintt~om of barlour quanrltles

(Y- REDNUI/AV

F i g r e 6, Line profiles for linear-velocity law and Gaussian scattering function.

- 5 0 + 5 +10

Figure 7. lntegrn~rd 11nc proiilrs for the two phase funct~ons

which, if convergent, would represent the N-solution of the problem. The convergence of the series solutions involving repeated operators was investigated, and the region of convergence delimited for atmospheres of arbitrary optical thickness for various values of Ro and S2, the maximum fractional perturbation in ll(r). This region is found to be the larger, the smaller the atmospheric thickness. The curves of conwrgsnce are shown in figures 8 and 9 and the error analysis is shown in flgure 10. They have extended this technlqur to Rayleigh phase matrix [39, 401 and tabulated scattering functions for an imperfect Rayleigh scattering semi-inf~nite atmosphere [43].

7he nonlinear singular integral equatlon for the H-function of an inhornogeneous plane-parallel atmosphere of arbitrary stratification is linearized by using the perturbation method developed in paper I . The N-solution of this equation is given. The iteration procedure for computing the fractional perturbation h(t ; p) in H(T; p) is also described. An error analysis is performed, and the relative error in the solution, caused by the neglect of the quadratic term, is computed. Curves showing the maximum relative errors in h(r; p ) and H(T; p ) as functions for the relative perturbation, s, were estimated. These curves will be useful for estimating the errors to be expected in any particular problem. For illustrating the method, H-functions for semi-infinite homogeneous atmospheres with = 0.300, 0.700, 0.925 and 0.975 are computed from the H-function for an atmosphere with f2 = 0.900.

Figure 8. The curves fio(S2) whch l lmt the dornaln of conterpnce of the solut~on for \arlous opt~cal depth\

Sz- Figure 9. Region of convergence of the h e a r solution for Rayleigh phase matnv

In order to describe the transfer of partially polarized radiation through an inhomogeneous semi-infinite atmosphere, they have introduced a matrix N-function by considering the limit of the K-functlon as the optical thickness of a finite atmosphere becomes infinite. The non-linear singular integral equation satisfied by N, generalizes the scalar H-equation of Sobolev for the fields in which the state of polarization must be taken Into account. This equation is solved by the perturbation method in the form of a Neumann series.

S- jc

FilYn 10. Maximum relative error in H ( M ) rn a function of for v.nou. of o (dashed) and of " = "' + ') (''lid HaIWhd a w . . regions where the lnmr solution not convergM (note the different scale of absclm in t k top ,jkmm).

A b h ~ & r has studied the scattering of risible radiation a m ~ n d the phe.ial a t m o s ~ h e ~ of Venus and concluded that the main contribution to & i , s that the b~@b of nus at inferior ~onjllnction E O n l S from matter which mtkn one order of

moE ~ f ~ i e n t l ~ i. the forGad direction than a Rayleigh u f i e m r *,,

considering the variation of extinction coefficients wlth waveiength and \.ariation of efficiency factor with colour, he concluded that these particles could be water particles 1261.

His collaboration with one of hlb students R . K. Bfiatia on planr:ar> atmo\phercs produced many interesting results [64, 69, 77, 78, 79, 80, 83. 88. 891. I t uould be difficult to narrate all these results here. The interested reader is referred to the orlglnai papers ior details. They calculated the absorption and polarization llne proflies and the cur\es of growth in the Integrated light of a planet o\er a large range of phabe angles assuming a semi-lnfmite atmosphece with Rayleigh phase matrix (see f~gures 1 1 and 12).

Figure 11. Polarizat~on line profiles at d~fferent angles (& = 0.2).

Abhyankar is an example of determination and commitment and a paragon of scientific virtue. He maintains scientifically inclined relationship with his colleagues, and others who are known to him through scientific research. He is uniformly warm towards all his students.

ALL 10 9900

TOTAL FLUX F C I I

Figure 12. Line profiles for the tot~+l flux F ~n the case of a strong line w~th = 0.2 at different phase angles.

.4fter retirement he has been appointed UGC Emeritus Professor. He is full of ideas and programmes and engaged in research as actively as he was used to earlier. Now that he is free from University administrtive duties, we can expect much more from him in the future.

We all wish a healthy and happy tirn: for Mrs Abhyankar who is always so pleasant to talk to and Professor Abhyankar.

References

1 Das, A R. & Abhyankar, K. D (1955) Nature 172, 496 Difference of temperature between pole and equator of the sun.

2. Das, A. K. & Abhyankar, K. D. (1955) V~sras Astr. 2,458. Temperature at the poles and the equator of the sun.

3. Abhyankar, K D. & Rarnanathan, A. S. (1955) Ap. J. 121, 739. Equivalent widths of atormc lines In sunspot spectra.

4. Struve, 0. & Abhyankar, K. D. (1955) Ap. J . 122, 409. The spectrum of Nu Eridani.

5 Abhyankar, K D (1957) Puhl ojrr Put!/ic 69. 385 Pcrlod and 9pectrum o i Plnrhett's btar 6 Stl-me. 0 , Sahdde. J & 4bhqdnhrr. K U (19jX) Ifrtlc Sot K Jc Lrzqr Sc'r 4. 70, 108 The bpectrum

of Plaskett's \tar 7 Abhydnkar. K D & Hyron Splnrdd (1958) Plrhl alrr Sac Pu~~lrc . 70. 41 1 Llgl~t varlablr~t\. of H D

47 129 8 Abh~ankar . K D (1959) .4p J Srcpp 4. 157 A rtudg of some earl1 t)pt . close binary srars Ph D thrslr 3 Ahhyanhar. K D (1959) 43rr J 64. 163 Stablllt). of strd~ght-11n.l solutions ~n ~ h c rrslrlcted problem 01

three hodlcr 10 Stableford. C & Abhydnkar, K D (1959) Jp J 130. 811 4 ~pecrrophotsnetric srud! of sebera! k t a

C a n ~ s malorla vdrl,ibles I I Abhyanhar, K D (1959) 4p J 130. 834 A D CMI-A new ultra hhort-per~od bailable 12 AbhyanKar. K D (1960) Obset~rur t~ 80. 26 A note on the e$olutlon of close blnar! ,)stems I3 Abhydnkar. K D (1962) 2 Ap 54, 25 Photometr~c elements of YY CMI 14 Abhyankar. K. D (1962) Nrzanlrah Oh3 Conrr 1. I Photornetr~c ob>rr\at~ons of YY C411. 15 Abhyanhar, K D (1963) J 0~111ania L't~rt ' Jcrence 1. I Close blnary stars-.\ re\lew 16 Abhyankar. K D (1963) Obtervarorv 83. 252 On the meaaurrmrnt of radial \elocltles from htgh

d~sperslon coude spectrograms 17 Abhyanhar, K D (1964) ,Vizamrah Obs Conrr 2. 1 Rad~al \elocltles of 63 G and K t) pe stars. 18 Abhyankar. K D (1964) J R asrr. So( Canoda 58, 118 Modern Astronomy In lndla 19 Abhyankar, K D (1964) 4p J 190. 1353. On the Schu\ter problem for a mo\lng atmosphere 1 20 Abhyankar. K D (1964) Ap J 140. 1353 On the Schuster problem for a moving atmosphere I1 21 Abhyanhar. K. D (1965) Ap J 141, 1056 On the Schuster problem for a moblng atmosphere 111 22 Abhyanhar, K I) & M B K Sarma (1966) .M Y. R .4.S 133, 437 A \tudy of the spectroscop~c blnaq [

CrB A 23 Abhyankar. K D (1965) Proc: TlFR tonf on cosrnrc ravs and mtrophr.s~cr, p. 782 Absorption lines In

moving stellar envelopes 24 Abhyankdr, K D (1967) Modern astrophys~cs, A memor~al to Otto Struve (ed. M Hack), Gordon and

~reach. 'p 199. Center-l~mb varlatlon of llne profile In a moving atmosphere. 25 ~ b h ~ $ n k a r , K D (1963) J P L Space Progra,nnre Sunzmarv 37-51, 111, 181. Scatterlng In the twlllght

atmospheres of Venus. 26 Abhyankar. K D (1968) 1car.u~ 9, 507. Scatterlng of vlslble radlatlon around the spherical atmosphere of

Venus 27 Abhyankar, K. D. & A L Fyrnat (1969) J P L Space Programme Summary 37-56.111.50. L~nearizatlon of

the global eqpatlons of radlat~ve transfer 28 Fyrnat, A L. & Abhyankar, K. D. (1969) JPL Space Programme Surnmarv 37-56, 111, 53 lteratlon

scheme for solv~ng the :inearlzed X- and H-equat~on 29 Abhyankar. K D & Fymat, A L. (1969) J Morh & Merh. 19, 115 A new type of multidimens~onai

stngular h e a r tntegral equation.

30 Fyrnat, A L. & Abhyankar, K. D (1969) J P L Space Programme Summary 37-57, 111, 57. Effect of llnearuat~on of the transfer problem on the computation of emergent radlatlon field.

31. Abhyankar. K D. & Fymat, A L (1969) 3PL Space Programme Summarv 37-57,111, 25 Error analys~s of the hnearzatlon method In r a d ~ a t ~ v e transfer theory.

32. Fyrnat, A. L. & Abhyankar, K. D. (1969) Ap. J. 158, 315 Theory of radiative transfer in lnhomogeneous atmospheres-I. Perturbation method.

33. Fymat, A. L & Abhyankar, K. D. (1969) Ap. J. 158. 3 i5 Theory of radiative transfer in lnhomogeneous atmospheres-11'Appl~catton of the perturbation method to a sernl-~nfin~te atmosphere.

34. Abhya~~kar , K. 5. & Fymat, A L. (1969) J. Marh. Phys. 10 1935. Relations between the elements of the phase matrix for scattering.

35. Abhyankar, K.. D. & Fymat, A. L. (1969) 3. qumri t Specrroc. radia. transfer 9, 1563. O n the solut~on HI (p) of Arnbarzurman-Chandrasekhar H-equatlon.

36. Abhyankar, K. D. & Fymat, A. L. (1970) Asrr. Ap. 4, 101. Imperfect Rayleigh scattering In semi-infintte. 37. Fymaf A. L. & Ahyankar, K. D (1970) J. quantrr. Spectrosc. radia. transfePl0.49. An alternate form of

Ueno's X- and Yequations for tnhornogeneous atmospheres. 38. Fymat, A. L. & Abhyankar, K. D. (1970) J. Marh. Mech 119, 587. A novel type of rnatrvr

mult~dimens~onal stngular linear integral equation.

jg Abhvankar. K D B; F>mnt, 4 L (IY70) 4p. J 159. 1009. Theory of radlatlve transfer ~n ~nhomo~eneous srmozphcrcl 111 Extcnsl~n oi the pertllrbatlon method to azimuth-Independent terms

40 ..\bhlsniar, li L) B Flrnat A L (1970) Ap J 159, 1019. Theory of radlatlve transfer In ~nhomogeneous atmospheres I \ AppIlca[ion of the matrix perturbation method to a serm-infinlte atmosphere

I! FymAt. A L & Xbh~ankor. K D (1970) .4ppl Opr 1075 An interferometrlc approach to the mcdsurernent of optlcal pokdrlzation

42 F!mat. .4 L Bi 4bhjanknr. li D (1971) J Gzoplr)'s Res 76, 732. Effect of molecular absorption on xatterlng b> planetar? armoapheres

43 Abh)ankdr, K D & Fymat, 4 L (1971) Ap J Suppl 195. 35 Tables of auxlllary functions for non- clin\ervatlcr Raylrlgh phase matrlx In the case of seml-lnfinite atmospheres

.W 4bhyanLar. K D (1972) J ~ n d / i r o PrarIeJh Acad SCI 9, 9 Welghlng the stars and the unlverse 45 4bh)ankar. K D ( 1972) Pro' :>nl/J un S P E C ~ ~ U T L O P I C szudies of asrrophys. mteresr, Osrnanla Unlverslty,

Hyderabad Potentlallt~es of Four~er spectroscopy 46 Sanwal. h . B. Parthasarsthy, M Abh~ankar, K D (1973) Observaror~~ 93, 30. UBV photometery of zeta

.4ung~e during the 1971-72 47 S a n ~ a l , K B . Sarma. M B. K , Panhasarathy, M . & Abhyankar, K. D (1974). Astr. & Ap. Supp. Ser

13, 81 BD-18O 3437, a new short period eclipsing blnary 48 Abhyankar, K . D., Parthasarathy, M & Sanwal, h' B. (1974) Asrr. Ap. Supp. 13, 101. UBV photometry

of RV Cor t~ . 19 Abhyankar. K D (1974) Pramano 2, 336. Varlatlon of the galactic force law in the reglon of the sun 50 Abhyankar, K. D. (1974) Bull. asrr Soc. India 2, 13. Variable x-ray sources and black holes in binarles 51 Abhyankar, K D. (1974) Bull. asrr. Soc. lnd~a 2, 14. Next decade in the study of solar and stellar

atmospheres 52 Abhyankar, K. D (1976) J. Andhra Prodosh Acad. SCI. 14, 71 Comets. 53 Abhyankar, K. D. (1976) Sfi. Rep. 13, 424. On the time of vernal equinox 54 Abhyankar, K D. (1975) Proc. solm phgs symp. held ar solar observarory. Udaipur 5 , 40. On the solar

atmospher~c model. 55 Kulkarnl, A G & Abhyankar, K. t) (t977). Bull. asrr. Soc. .his 5, 105. Light elements of TT Hydrae 56. Abhyankar, K. D , Sarma, M. B. K. & Kulkarni, A. G. (1978) b n t r i from Nizanuah and Japal-

Rangapur Observatories, No. 5. Phorometnc Observations of WX Eridanl. 57 Kulkarm, A G & Abhyankar, K. D. (1978) Contr from Nizamlah and Japal-Rangapur Observatories,

No. 9, Photometric observations of TT Hydrae. 58 Sarma, M. 8 . K. & Abhyankar, K. D. (1979) Asrr. Sp. Sci. 65, 443 Photometric elements of W X Eridani;

lAL/ Coll. No 46, 1978 59 Kulkaml, A. G. & Abhyankar, K. D. (1980) Asrr. Sp. Sci. 67,205. Absolute dimensions of IT Hydrae. 60. Panchatsaram, T. k Abhyankar, K. D. (1981) Bull. asrr. Soc. of India 9,31 SW Lacertae-A quadruple

system. 61 Panchatsaram, T. % Abhyankar, K. D. (1981) J. Ap. Asrr. 2, 29 An analysis of the orbital periods of

some ecllpslag binaries suspected to be in the pre-maln sequence coniraction phase of tvolutlon 62 ICulkami, A G . 8t Abhyankar, K. D. (1981) J. Ap. Asrr. 2, 1 19. A'modei of the algol type close blnary 'IT

Hydrat. 63 Abhyankar, K. D. (1981) Bull. mtr. Soc. Indza. 9, 99. Evolut~on and period changes in binary stars. 64. Bhaba, R. K. & Abhyankar, K. D. (1981) Bull. astr. Soc. In& 9, 181. Role of scattering in emergent

planetary radiation. 65. Panchatsaram, T. & Abhyankar, K. D. (1981). Bull. ostr. Soc. b~dia 9, 243. A note on the orbital

pcnods of the eclipsing binaries CM Lac, AB and YY Eri. 66. Anthony Ra~u, K. & Abhyanhr, K . D. (1982) Proc. Indian Nar. Sci. Acad. 48, Suppl. No 3, 91 Two

colour photometry of the solar corona. 67. ~ b h y m k a r , K D. & Subrahmnyam, P. V. (1982). Proc. Indian Nat. Sci. Acod. 48, Suppl. No. 3, 55

Effect of secing plus scattcnng on the observed intensity distribution in the corona. 68. Panchatsararn, T. & Abhyankar, K. D. (1982) In IAU CON. No. 69, p. 47. A study of the orbital periods

of 22 eclipsing binaries. 69. Bhatia, R. K. & Abhyankar, K. D. (1982). Proc. Indo-US workshop on high resolu~~on spectroscopy at

BARC. Bombay. On the phase variation of absorption bands in Venus atmosphere. 70. Kaul, J. 8r Abhyanlrar, K. D. (1982) J. Ap. Asrr. 3, 93. Analysis of the UBV light curves of TT Hya by

Kopal's frequency domaln method.

71 Panchatsaram, T & ,Abhyankar. K D ( 19x2) Bull clrrr So, Ititlrrl 10 1 I i 4s~ul.i: pc.rlr,c L n . i Q p .:. :t.n ecl1p51ng blnarle,

72 Abhyankar. 1C 1) ( 1982) Sot Scr 5, i b 4stront>rn:, and AIIIUIO~! 73 i\hhknnhar li 1) (1982) Curr .%I 51, 955 Three and h ~ l l ccniurisr of hinark bldr rr$t.arch 74 Xhhlnnhar. li 1) r 19x2) 10 l'zur Hooh 451,- Sol Ir~Jlu 1 Gsnesls dnd grobt'n C I I dstr~)nom:,.i; s,>~lct! ( I '

1nd1.i 75 4hh)dnLar. I( I) & Panchatsaram. 1- ( I ' % ? ) Bull utrr Soc Itlllru 10, 3 1 5 1 ~ g h t - r ~ m c cffcct :r \ h H I :

and E R Ori 76 AShyanLar, K D . Rajasekhara Rao. N & Lohanadharn. B (19F2) Bull urlr Sc~c l t idu 10 323

Photometric solllt~on of the ecllps~ng h~nary TT 4ur1gat. 7" Bhatia. R K . & Abhyankar. IC D (1987) J Ap 47rr 3 303 Intrn\lt) and polarlzat~,m lint- prol~:r\ :n u

<em:-~nf~nlte Rayle~gh scatteitng planetary atn?osphrre I Integrated flux 78 Bhatla. R K . & Ahhyankar. K D (1983) J Ap A J I ~ 4 77 Iqtrns~t) and polurlrdtlon linc prol~tt.5 :a, -I

s e m l - ~ n f ~ n ~ t e Raylelgh scatterlng planetar) atrnosphero 11 Varlatlon of equl\alent \sldth w c r tht. d.\L 79 Bhat~a. R K & AhhyanLar, K D (1983) J Ap .4\rr 4 35 Intsnslt! rnd polarl/ar~ot. I ~ n c prolile\ :n .I

s r m ~ - ~ n f ~ n l t c Rayle~gh scatterlng planetar). atmospheres--Ill Vdrlar~on ol p o l a r ~ r a t ~ o n prol~le\ and thc stoke\ parameter Q over the d ~ s h

80 Bhatca, f: K & Abhyankar. K D (198j) . 4 p Sp Str 96 107 Mulr~ple scatterlng and phase variation of equlvalent w~dths for phase funct~ons of tkpe P(cos 8) = wo + w , PI (cos 8) + w! P: (cos 8)

81 Radhahrishnan. K R , Sarma. M B K 8 Abhyankar, K. D I i984) .4p Sp S1.r 99. 219 Photnnxt r~c and spectroscopic study of R CMa

82 Abhyankar, K D (1984) A p Sp. Scr 99. 355 Mass loss In sem~detached Blnarles 83 Abhyankar, K. D & ~ h a i r a , R K (1984) G r r h ,%toon Planer, 30, 175 4 recons~derat~on of the rffectl\r

depth of llne forrnat~on in planetar) atmospheres 84 Radhahr~shnan. K R . Abhyankar, K D & Sarma. M B K (1984) Butt usrr Soc ftlrl~u 12. I82 Perlnd

study of algol type echpslng blnary - R C a n ~ s majorls 85. Abhyankar, K D & Subrahmanyarn, P V (1984) Bull asrr So,, I n l a 12, 190. Correct~on for the

observed lntenslty d~stribution In solar corona 86 Abhyankar, K D (1984) Plarrnurll Juhrke Syrnp Nr:mrlah O h 3 on b ~ n o r , and t?~ulr~p le srsre17u'. 1

Mult~pllclty among ecl~pslng binar~es 87. Abhyankar, K D & Panchatsaram, T (1984). M. N R A S 211. 75 Light t!me effect In T W Draconis 88. Bhat~a , R K. & Abhyankar, K D (1984) Bull asrr Soc lrrdra 12, 384 Multlpie scattering and

dependence of phase varlatlon of equlvalent wldths o n llne profiie 89. Bhatia, R. K & Abhyankar. K D (1984) Bull. asir. Soc incha 12, 364. H- and auxiliary funct~ons for

phase functrons of the type P(cos 0) = w + U, Pl(cos 8) + w P2 (cos 8). 90. Radhakrlshnan, K. R. & Abhyankar, K D. ( 1984) Bull. asrr Joe Jnhg 12.41 1. Spectroscopic orbit of R

CMa. 91. Radhakrlshnan, K R., Sarma, M. B. K. & Abhyankar, K. D (1985) Bull. asrr Sot. India 13, 261. UBV

photometric study of R CMa. 92. Abhyankar, K D. & Bhatia, R. K (1985) Bull asrr Soc lndra 13, 275. O n the stab~litg of the

hypothet~cal quintet system of TW Draconis. 93 . Sarma, C . V . S. R , Sarrna, M B. K & Abhyankar, K. D (1985) Bull. asrr Soc Indra 13. 346 Pertod

study of the RS CVn eclipsing blnary SV Camelopardalis. 94. Praveen Nagar & Abhyankar, K D. (1986) In IAU Coll. No. 87. Hydrogen defic~ent stars and related

objects (eds K. Hunger er at.), 251. Hydrogen delictency In algol secondaries. 95. Anthony Raju, K . & Abhyankar, K. D. (1986) Bull. aslr. Soc. India 14, 217. Polarlzatlon and electron

densities in the solar corona of 1980 February 16