The standard systems of stellar rotation measurements (II)

Chin. As&on. Astrophys. (lSS3)17/4, A translation of Acta Astron. Sin. (lSS3)34/2,15S-164

@ Pergamon Press Ltd Printed in Great Britain 0275-1062/93$24.00+.00

393-400

The standard systems of stellar rotation measurements (II) t PAN Yunnan Observatory,

TAN

Kai-ke Chinese

Academy

Hui-song of Sciences,

Kunming

650011

Abstract The limb darkening effect on the measurement of stellar rotation is discussed in this paper. It is shown that this effect plays an important role in the measurement of V. sin i. In the extreme case with the limb darkening coefficient of 1.0, it may cause a difference up to 17%. As a sequel to paper [l], this work presents the following new explanation for the systematic differences between Slettebak’s new and old systems. The main causes of the systematic differences are: (1) The old system made an inadequate twice repeated correction for the limb darkening. (2) Owing to the historical reason, the old system used too large limb darkening coefficients. Key

words:

stellar

rotation

- spectral

lines - limb darkening

1. INTRODUCTION In the measurement and study of stellar rotation, the standard systems are of very important significance. However, between the two existing systems, i.e., Slettebak’s old and new systems, there are quite conspicuous systematic differences11*21, which can be summarized in the following. differences between the two systems 1. When V, sin i < 30kms-‘ , no evident systematic can be found. 2. For the B-type stars with V, sin i < 5Okms-’ , the measured values in the old system are on the average 17.9% larger than those in the new system. 3. For the A- and F-type stars with V, sin i < 50kms-‘, the values in the old system are on the average 8.5% larger than those in the new system. Slettebak et a1.131and CoIlingI intuitively suggested that the systematic differences are caused by the distinction between the models used in the two systems and also by the discrepancies in observational techniques. in paper [l], these factors cannot account for the systematic According to our investigationI’ differences mentioned above. differences. Therefore,

At least, they are not the main reasons of the occurrence of such continuing the work in paper [l], we will discuss the limb darkening

t Supported by National Natural Science Foundation Received 1992 February 12; revised version November 9

PAN Kai-ke & TAN H&song

394

effect on the measurement results. This may provide systems.

2. THE

of stellar rotation a new explanation

INFLUENCE

OF THE

and interpret the before-mentioned statistical for the systematic differences between the two

LIMB

DARKENING

EFFECT

Since the fifties of this century, there have been already several papersl’-“1 which claimed that the limb darkening effect on the stellar rotation measurements could be negligible. Because no detailed discussion has been done, it is worth to perform a further investigation of this question. According to the work of Graylgl, the radiation flux profile F(A) of a rotating star can be expressed as the convolution of the eigenprofile H(A) o f a nonrotating star and the profile produced by rotation G(AX), i.e., F(A) = H(A) * G(AA),

(1)

where 2(1 -

E) [l -

G(At)

-

A%L -

LV,sin i/c,

(A~/AIL)zl~ + f SE[ 1 - (AZ/AI~)Z] xA1~(1 -

(2)

e/3)

I Here X is the wavelength of the line center, c is the light velocity, E is the limb darkening coefficient, and AA is the Doppler displacement caused by rotation. As inferred from Eqs.(l) and (2), G(AA) is mainly determined by the rotational velocity of star V, sin i. The larger is V, sin i, the larger is its half width. Therefore, when V, sin i is large enough, or when H(X) can be taken to be the 6 -function, the influence of H(J) on half width of F(A) may be neglected. Then the half width of the radiation flux profile rotating stars can be expressed with the half width of the rotational profile. For the convenience of discussion, Eq.(2) may be written as c(az)

-

cl[i -

(AZ/AL,)+

2(1 - E) AU 1 -E/3)

=I -

4

cI[i -

(AI/A~,)*],

’

(3)

s

ca -

2AIL( 1 - E/3). So the peak value of G(Ax) is

G(A~)max=

G(0)

Let AXh be the Doppler +

GW),,

-

=

cl

(4

+ c2.

displacement c,t 1 -

corresponding

(AL,/AI~)+

to $G(AA),,,

+ cJ1 -

(AblAL)‘].

then (5)

By the definition of the full width at half-maximum (FWHM) of a spectral line, the FWHM of G(AA) is 2aAXh. Solving simultaneously the above equations, we get the relation between the FWHM of the rotational profile and V, sin i : V,&i

-

E X FWHM/ZI(l

-

yr)j,

Stellar Rotation Measurements

-2(I

-s)+

1

4(1-ssy

++s

Y’

[

395

f

2(1-a)++6

31

(6)

ti

Substituting E:= 0.0 and 1.0 into (5) and (6) respectively, we obtain the relations between the FWHMs of G(AX) and V, sini’s in these two extreme cases: V, sin i = c x FWHM/1.73205 A, (E = O.O),

(7)

V, sin i I= c x FWHM/1.41421 X, (E = l.O),

(8)

From this it can be readily seen that in the extreme cases the eigenprofile may to be the &function, or that V, sin i is so large that the influence of H(A) on the line profiles of rotating stars can be neglected and the limb darkening coefficient in 1.0. Then the influence of the limb darkening effect on the measured value of stellar can obtain: lY,sln&_,.,f?‘,sinij,rp.p 1V,sinL,.,

_

1.73205 - 1.41421 1.73205

~ r8.3qb.

be taken spectral Eq.(3) is rotation

(9)

In reality, the influence of the limb darkening effect on the measured value of stellar rotation becomes larger with the increasing E, so it is a function of the wavelength of the spectral line, the spectral type of the star, etc. For the line Fe1 X 4476 of F2 stars, the limb darkening coefficient is 0.761l”~lll. By a derivation similar to that of Eq.(9), the influence of the limb darkening effect on the measured value of V, sin i is 13.1%. Because the eigenprofile cannot be an intrinsic 6 -function, the values 18.3% and 13.1% obtained here must be considered to be the upper limits of the influence of limb darkening on the measurement of stellar rotation for specific conditions of E, For the discussion of the influence of the limb darkening efIect including the influence of the eigenprofile, we observed the profile of Fe1 A 4476 of HD 128167 (spectral type F2) with the Coude-CCD system of the Kitt Peak Observatory, used it as the eigenprofile (the FWHM of this line is 0.36A) and calculated the profile broadened by rotation by means of convolution with the rotational profile G(AA). Taking E = 0.0,0.76 and V, sin i to be, respectively, 50, 60,70,80, 90, 100, 110, 120, 126 km s-i, we calculated two series of profiles broadened by rotation, measured their FWHMs and obtained the following relations: V, sin i -

44.27(-I- 0.23) X FWHM -l- 2.9( + 0.5),

(a -

6.76).

(IO)

Y,sini-

37.90(+0.31)

(8 -

6.61).

(II)

X FWHM C 5.4(-t&g),

From these two equations we can readily see that the influence of the limb darkening effect on the measured values of stellar rotation depends on the V, sin i itself. For F2 stars, when V, sin i varies from 50 km s-* to 126 km s-l, its influence changes from 8.5% to 12.1%. The value 126 km s-l used here is the maximum rotational velocity chosen for F2V stars by the modell’2l of Slettebak’s new system. The component of angular vefocity corresponding to this V, sin i, i.e., the ratio of the rotational angular velocity to the breaking critical angular velocity, it merely 0.5.

396

PAN Kai-ke & TAN H&song

Table 1

sp. of

The Limb DarkeningCoefficientsused by the Old System of S1etteba.k

Lines

Star

He He He He

08-BZ

I-

Limb-darkening of

I 4026 1 4471 II 4200 I 4686

B2-B8

He 1 4471

BB--A2

Mg II 4481 He 1 4471

the

‘Old’

System

0.67 Slettebak:

(see

Coefficients

1956)

of Gryger

et al

0.37-0.43 0.34-0.39 0.36-0.41 0.33-0.38 0.39-0.51

0.73 (Slettebak:

0.51-0.66

1954)

A3-A7

MgII

4481

0.73

(Sletrebak:

1955)

m--F2

MgII

4481

0.76

(Slettebak:

1955)

0.70-0.77

FO-FS

Fe1 4071

0.82

(Slettebak:

1955)

0.82-0.84

F2-F8

Fe1 4071

0.84

(Slcttebak:

1955)

0.83-0.85

F7-G5

Fe1 4071

0.87

(Slettebak:

1955)

0.85-0.90

0.67-0.74

In order to roughly

estimate the of the limb darkening effect on the measured values of of the eigenprofile to be included and for E = 1.0, we took E = 1.0 instead of 0.76 in the derivation of the relation (10) and made similar calculations. Then it follows that when V, sin i changes from 50 to 126 km s-l, the influence of the limb darkening effect varies from 11.5% to 16.2%. Because 126 km s-l is merely the value of V, sin i for the component of the angular rotational velocity equal to 0.5, in combination with the previously obtained upper limit 18.3% we estimate that the influence of the limb darkening effect on the measured value of stellar rotational velocity may possibly attain 17%.

V, sin i with the influence

3. MAIN

CAUSES THE

3.1 Selection

OF THE

SYSTEMATIC

SLETTEBAK

NEW

of Limb Darkening

AND

DIFFERENCES OLD

BETWEEN

SYSTEMS

Coefficient

It is undoubted that for his old system

established

in the fifties Slettebak

chose from the

literature of that time the most reasonable limb darkening coefficient. However, with the increasing accuracy of measurements and with the improved atmospheric model used for the amputation of the limb darkening coefficient, the value of the coefficient adopted by the old system differs to certain degrees with those provided by the later papers (10, 11, 131. Table 1 gives the coefficients of limb darkening used by the old system. We are aware of the fact that the atmospheric model used by Grygar et a1.[131 is quite similar to that of Slettebak’s new system. As a comparison, the limb darkening coefficients calculated by them are also listed in Table 1. It can be seen from Table 1 that for stars with spectral type-s earlier than A2 the limb darkening coefficients used by the old system are larger than Grygar et al.‘s corresponding values. Because the atmospheric model of the new system is close to that used by Grygar et al., we shall investigate the systematic differences caused by various limb darkening coefficients in the measurement of V, sin i.

397

Stellar Rotation Measurements

By substitution obtained:

of various values of E into Eq.(6),

the following

different

relations

can be

V, sini = c x FWHM/1.6395A,

(E = 0.35).

(12)

V, sin i = c x FWHM/1.6085&

(E = 0.45).

(13)

V, sin i = c x FWHM/1.5757X,

(E = 0.55).

(14)

K sin i = c x FWHM/1.5345&

(E = 0.67).

(15)

V, sini

(E = 0.73).

(16)

= c x FWHM/1.5139A,

As shown by Table 1, for the spectral line He1 X4471 of the early B-type stars the E chosen by the old system is 0.67, but the values computed by Grygar et al. lie between 0.34 difference possibly caused and 0.45. According to Eqs.(l2) and (15), the extreme systematic by these various E’S is [~,si,.xi],,~.~

-

[v,sinilr-od

[V,siniL-0.67

_

1.6395 - 1.5345 1.6395

6.4%.

By the same reasoning, similar systematic differences exist for other He1 X4471 of the late B-type stars, the corresponding value is

(17) stars.

For the line

(18) For the line MgII A4481 of the A-type

[V,

stars earlier

sin&D.,I - [ v, sini]c~‘O.(r --&1.3%. 1v, sixlL0.7,

than A2, the systematic

difference

is

(19)

For the late A- and F-type stars, the E’S used by the old system basically agree with Therefore, the systematic differences caused by an Grygar et al.‘s[13l computed values. inadequate choice of E can be ignored. Because the accuracy of measurements of stellar rotation at that time is merely about lo%, the significance of Eqs.( 17)-( 19) is quite trivial for individual measurements. However, if a certain factor has intrinsic influence on the measured value of V, sin i, even if the amount is less than lo%, its trend should be shown in the statistical studies like this work. Hence we think that one of the reasons of the existence of systematic differences between the new and the old systems is that the old system used too large limb darkening coefficients. 3.2 The Twice Repeated Correction of the Limb Darkening In Slettebak et al.‘s work of old system[5*14-171, th ey adopted the observed profiles of sharp-line stars as the eigenprofiles of nonrotating stars, utilized a diagrammatic method to calculate the radiation flux profiles of the stars to be measured and then got the rotational velocity of the stars by comparison of the computed and measured profiles. Since the influence of the limb darkening effect has already been embodied in the observed profiles of sharp-line stars and this fact was overlooked in the correction for the limb darkening in

PAN Kai-ke & TAN Hui-song

398

the old system, so the rotational profiles calculated for the old system contain the twice repeated correction of the limb darkening. We sincerely thank Prof. Arne Slettebak of the Ohio State University for the helpful correspondence about this question [11p201.Due to the twice repeated correction of the limb darkening,

the spectral

line profiles of rotating

stars are inadequately

changed

to be

H(1)(1-s+sCoSe)r,cosedl0 F(L)

-

-

H(i)IG'(Al).

(20)

1,cosedla Equation (20) is formally still the convolution of the eigenprofile and the rotational profile. But the rotational profile has already been altered to be

I!( 1 - s -I- s cos 8)‘dyl

R

G'(AJ) f:(l- s f where R is the radius leads to

G'(Al)-

of the star and yr = R[l -

*A&(

1

1 __,3j_

+ 7cs(l -

(20

s)ll -

Because E is concerned with the value of rotation thus caused also limb darkening coefficients used in that of Eq.(6), we get the following

t

scost?)cosBdw

-

(21)

(Ah/Ax,)2]l/2, The integration

8171

-

(A~/AL,>~]

of Eq.(21)

(AL/A~,)~ + +1

-

(AI/Ah)‘1 #.

(22)

spectral type, the amplitude of increase of the measured changes with the spectral type. With the values of the the old system (Table 1) and by a derivation similar to relations between the FWHMs of G’(AA) and V, sin i’s:

V, sin i = c x FWHM/1.3732&

(E = 0.67).

(23)

V, sin i = c x FWHM/l.3430~,

(E = 0.73).

(24)

By comparison of (15) and (23) as well as (16) and (24), it can be seen that owing to the twice repeated correction of the limb darkening the measured values of Ve sini are erroneously exaggerated. For early B-type stars, the amplitude of exaggeration is (1.5345 - 1.3732)/‘1.5345 For late B-, A- and early F-type (1.5139-

1.3430)/X5139

= 10.5%. stars,

the amplitude

= 11.3%.

(25) is (26)

Stellar Rotation Measurements

399

Now we can interpret the statistical results stated in the first section. For the B-type stars with comparatively large V,sini the twice repeated correction of limb darkening in the old system leads to an inadequate increase of V, sini by 10.5%-11.3%. Besides, the exaggerated limb darkening coefficient further enlarges the measured values by 4.6%-6.4%. So the V,sini’s of this system should be 16%-17% larger than those of the new system. This agrees rather well with the statistical conclusion (2). For the A- and F-type stars with large V, sin i’s the twice repeated correction of limb darkening in the old system gives rise to an increase of the measured values by 11.3%, and this result basically coincides with the statistical conclusion (3) of section 1. It follows from Eq.( 1) that the radiation flux profile of rotating stars can be expressed as the convolution of the eigenprofile H(X) and the rotational profile C(AX). When V, sin i is small, the radiation flux profile is mainly determined by H(X) an’ the two factors mentioned above can mtsrely affect G(AX). Therefore, when V, sin i < 30 km s-l, no evident systematic differences can be detected between the new and old systems. Thus the statistical conclusion (1) in the first section can be explained. Although the overwhelming majority of the systematic differences existing between the new and old systems can be interpreted with the two factors mentioned in this paper, still a small part has not been accounted for, especially for A- and F-type stars. Perhaps the distinction between the models of the two systems may provide an interpretation of the remaining portion of systematic differences. This question awaits further investigation. Summing up all the above, this paper has drawn the following conclusions: A: The effect of limb darkening plays an important role in the measurement of stellar rotational velocities. For the line Fe1 X4476 of F2 stars, its influence may attain 12%. For the extreme case with the limb darkening coefficient close to 1.0, the influence of this effect on the-measured values of V, sin i can amount to 17%. B: The systematic differences existing between Slettebak’s new and old systems are mainly due to the following reasons. (1) The old system made inadequate twice repeated correction of the limb darkening. (2) The old system used too large limb darkening coefficients. Thanks are due to the much support and help to this work ACKNOWLEDGEMENTS kindly offered by Dr. G. W. Collins II and Prof. A. Slettebak of the Ohio State University and Dr. A. H. Abt of the Kitt Peak Observatory. Thanks are also due to Mr. Wang Xun-hao of the Yunnan Observatory for useful suggestions. A part of the observational material used in this vyork was obtained by the authors with the Coudb Feed Telescope of the Kitt Peak Observatory. We are grateful to Dr. De Young, the director of KPNO, for the arrangement of observation time. This work is partly supported by the S. S. Huang Foundation.

References [l]

Pan Kai-kc, Tan H&song et al., AA&,

[2 ]

Vesugi A. and Fuducla I., Revised Catalague of Stellar Rotational Velocities, 1982

X+91,33,395

[3]

Slettebak A., Collins CX’W. H. et al., ApJS, 1975,29,

[ 4]

Collins G. W. H., private comgnunication, 1990

137

400

PAN Kai-ke & TAN Hui-song

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