Properties of Double Stars: A Survey of Parallaxes and Orbits [Reprint 2016 ed.] 9781512800364

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Properties of Double Stars

^Má/i-e'ta-

etc/

S^ó^ótcc

Plate I. Tailpiece of the 24-inch refractor at Sproul O b s e r v a t o r y used f o r p h o t o g r a p h i c parallax d e t e r m i n a t i o n . Notice the plate holder a n d the guiding eyepiece. (Courtesy Dr. P. van de K a m p . )

Properties of Double Stars A SURVEY OF PARALLAXES AND ORBITS

LEENDERT

BINNENDIJK

Professor of Astronomy University of Pennsylvania

Philadelphia UNIVERSITY OF PENNSYLVANIA PRESS

© 1960 by the Trustees of the University of Pennsylvania Published in Great Britain, India, and Pakistan by the Oxford University Press London, Bombay, and Karachi Library of Congress Catalogue Card Number: 58-8011

Printed in Great Britain by W. & J. Mackay & Co Ltd, Chatham

Preface THIS BOOK IS BASED ON LECTURES GIVEN BY THE AUTHOR IN

an advanced year course for students who had finished at least a descriptive course in astronomy and who had the necessary basic knowledge in physics and mathematics. The intention has been to give the student an understanding of the double star problem, beginning with the precautions one has to take even before observations are started and concluding with the final results of the orbital elements. As a consequence, considerable space is used to explain provisional solutions. The definitive solution is then described using the least squares method of the professional astronomer. In Chapters I, III, and V, the student will find an introduction to astrometry, spectroscopy, and photometry, respectively, as a preparation for the observational techniques and the reductions to be carried out. In the same chapters a rather complete summary of methods of parallax determination is given, because it is important to know the distance both for a binary and for a single star. Some other topics are only touched upon. These chapters can be read in this succession if desired. In Chapters Π, IV, and VI, the different methods of orbital determination which are still in use are discussed and fundamental properties, like mass, size, and density, are studied. The standard nomenclature has been followed as far as possible. After each chapter selected references are included. No completeness is intended here. As a rule the original publications are mentioned and those which give a summary of the subject or have an extensive bibliography. 5

Preface

6

It is a privilege to extend my sincere thanks to three astronomers who are experts in the three fields of double stars covered in this volume and who have given me very valuable assistance in the preparation of the manuscript. They are Dr. Peter van de Kamp of Swarthmore College for Chapters I and II, Dr. Dean B. McLaughlin of the University of Michigan for Chapters III and IV, Dr. F. Bradshaw Wood of the University of Pennsylvania for Chapters V and VI. It is a pleasure to express my gratitude to Dr. William Blitzstein, Dr. Robert H. Koch and Mrs. Beverly B. Bookmyer, all of the University of Pennsylvania, for their help during the reading of the proofs. L. University of Philadelphia

Pennsylvania

Binnendijk

Contents Preface page 5 I. ASTROMETRY 17 1. Three fundamental formulae of spherical trigonometry 17 2. Trigonometric parallax 21 3. Parallax in celestial longitude and latitude 23 4. Comparison with aberration 25 5. Parallax in right ascension and declination 25 6. Parallax factors 28 7. Proper motion and parallax 30 8. Photographic stellar parallaxes 32 9. Observation 34 10. Measurement 37 11. Dependences 37 12. Method of dependences 40 13. Plate solutions 42 14. Computation of dependences 44 15. Number of comparison stars 44 16. Relative and absolute trigonometric parallax 45 17. Proper motions 48 18. Solar motion and apex 48 19. Secular parallax 51 II. VISUAL DOUBLE STARS 20. Observation 7

56 56

8

Contents

21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40.

Orbital elements defined Gravitation law and Kepler's laws Apparent ellipse and true ellipse Method of Kowalsky and Glasenapp Method of Zwiers and Russell Method of Thiele and Innes Differential corrections Application to the line of sight Apparent orbit a straight line Interferometer Sum of the masses Mass-luminosity relation Dynamical parallax Astrometric double stars Resolved astrometric binary Photocentric orbit Unresolved astrometric binary Mass determination Résumé Proofs of certain formulae

III. SPECTROSCOPY 41. The Spectrograph 42. Observation and measurement 43. Reduction 44. Daily rotation 45. Yearly revolution 46. Heliocentric correction in α, δ 47. Variable radial velocity 48. Equivalent width 49. Curve of growth 50. Spectral classification in two dimensions 51. Spectroscopic parallax

page 59 61 62 63 69 72 81 81 83 85 88 89 91 92 93 94 97 99 100 100 106 106 108 109 113 114 120 123 124 126 128 132

Contents

9

52. Peculiar spectra page 136 53. Interstellar lines 138 54. Parallax from differential galactic rotation 138 IV. SPECTROSCOPIC DOUBLE STARS 55. One spectrum visible 56. Method of Lehmann-Filhés 57. Method of Schwarzschild and Zurhellen 58. Method of Wilsing and Russell 59. Method of Russell 60. Method of Laves and King 61. Mass function (one spectrum visible) 62. Rotation effect 63. Gas streams 64. Two spectra visible 65. Mass ratio (two spectra) 66. Influence of reflection effect 67. Ratio of intensities 68. Diameter of Cepheids 69. Period-luminosity relations

148 148 152 157 163 165 168 170 171 173 176 178 179 180 186 193

Y. PHOTOMETRY 70. Intensity 71. Magnitude 72. Color and reddening 73. Black body energy distribution 74. Observed energy distribution 75. Visual method 76. Photographic method 77. Objective grating 78. Rich star field 79. Measurement 80. Reduction

198 198 199 201 201 210 214 216 217 219 220 221

10

Contents

81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96.

Magnitude systems Discovery of variables Bright variable Faint variable Light time Period Photo-electric method Open clusters Extinction Light curve Polarization Moving cluster parallax Spectrum-magnitude diagram Color-magnitude diagram Open cluster parallax Globular clusters

VI. ECLIPSING VARIABLES 97. Units defined 98. Algol type 99. Loss of light 100. Depth relations 101. The relation between S/rg, k and α 102. Dynamical condition 103. Determination of k, rg, i with / f u n c t i o n (method I) 104. Differential corrections 105. Complete eclipses with Φ and φ functions (method II) 106. Incomplete eclipses with χ and q functions (method II) 107. Limb darkening 108. Influence of limb darkening

page 223 224 226 227 228 232 233 237 239 244 246 247 248 250 251 253 258 258 259 263 264 265 267 268 271 272 276 280 283

Contents

109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120. 121. 122. 123. 124. 125. 126. 127.

β Lyrae type Oblateness Dynamical condition for oblate stars Oblateness with limb darkening W Ursae Majoris type Kinds of eclipses Gravitation effect Gravitation effect and limb darkening Reflection effect Approximate rectification Reflection and reradiation Correct rectification Visible and invisible companions Résumé Eccentricity of the orbit Rotation of the line of apsides Position angle of the node Extended atmospheres Information about masses, sizes, densities Index of Authors Index of Subjects

11 page 288 290 293 298 304 307 308 312 313 316 317 320 324 325 326 330 334 335 337 343 345

Plates I Tailpiece of the 24-inch refractor at Sproul Observatory used for photographic parallax determination frontispiece II Illustration of the parallax and proper motion of Barnard's star facing page 32 III Micrometer of the 36-inch refractor at Lick Observatory facing page 33 IV Double star camera attached to the Dearborn 18£-inch refractor facing page 64 V Photographs of the visual binary system Krueger 60 facing page 65 VI The new spectrograph attached to the 72-inch reflecting telescope of the Dominion Astrophysical Observatory, Victoria facing page 192 VII Negative spectra of the spectroscopic double Mizar on ten different dates facing page 193 Vili The Pierce photo-electric photometer attached to the 15-inch horizontal telescope at Flower and Cook Observatory facing page 224 IX Photo-electric photometer attached to the 16-inch Goodsell refractor, now attached to the 28^-inch reflecting telescope at Flower and Cook Observatory facing page 225 13

Abbreviations American Association for the Advancement of Science Astronomical Journal Astronomische Nachrichten Astrophysical Journal Bulletin of the Astronomical Institutes of the Netherlands I.A.U. International Astronomical Union L.O.B. Lick Observatory Bulletin Μ.Ν. Monthly Notices of the Royal Astronomical Society Mount Wilson Observatory Mt. W. P. Α.S.Ρ. Publications of the Royal Astronomical Society of the Pacific Pop. Astr. Popular Astronomy R.A.S.C. Royal Astronomical Society of Canada Z.f Aph. Zeitschrift für Astrophysik A.A.A.S. A.J. A.N. Αρ. J. Β. Α.Ν.

14

Properties of Double Stars

I Astrometry ASTROMETRY

MEANS

POSITION

DETERMINATION

FOR

THE

purpose of deriving the proper motion, and parallax of a star, and in addition the orbital motion for a double star. J. Bradley's effort to obtain a measurable parallax led to the discovery of aberration and nutation, both of which are much larger shifts in stellar positions than the annual parallax. W. Herschel's attempt to measure parallax led to the discovery of the physical double stars. In 1838 the first parallaxes were measured with a meridian circle and with a heÜometer. Those were thus visual observations. Now we observe the parallax only by photographic means. However, in case of the orbital motion of double stars, both the visitai and photographic methods are used. 1. Threefimdamentalformulae of spherical trigonometry. Since astrometric measures are essentially measures of a star's position on the celestial sphere, we must consider first some of the fundamentals of spherical trigonometry. A spherical triangle is a part of the surface of a sphere bounded by three great circles. Both the angles and the sides of the spherical triangle are expressed in degrees. The derivation of the three fundamental formulae of spherical trigonometry follows : (1) In Figure 1 the center of the sphere is at O. The spherical triangle ABC has the sides a, b, c. The plane ADE is tangent 17

18

Properties of Double Stars

0

Figure 1. The spherical triangle ABC. The center of the sphere is at O. The plane ADE is tangent to the sphere at A.

to the sphere at A, and therefore Z. OAE = / . OAD = 90°. In Δ ADE and Δ ODE we express DE with the cosine rule (OA= 1) : DES = tan2ò + tan2c — 2 tan b tan c cos A DE2 = sec26 + sec2c — 2 sec b sec c cos a = 1 + tan2¿> + 1 + tan2c — 2 sec b sec c cos a Thus : — tan b tan c cos A = 1 — sec b sec c cos a — sin b sin c cos A = cos b cos c — cos a The result is the cosine rule, which we will write for all the three sides. cos a — cos b cos c + sin b sin c cos A "j cos b = cos c cos a + sin c sin a cos Β > (1) cos c = cos a cos b + sin a sin b cos C

J

When C = 90° we get: cos c = cos a cos b

(2)

Astrometry

19

(2) S q u a r e sin b sin c cos A = cos a — cos b cos c s i n s è sin 2 c c o s M = cos 2 a — 2 cos a cos b cos c + cos 2 6 c o s 2 c T h e left side o f t h e e q u a t i o n c a n b e w r i t t e n a s : sin 2 ô sin 2 c (1 — s i n M ) - sin 2 6 sin 2 c — sin 2 6 sin 2 c sin* sin 2 c s i n M = 1 — c o s *b — c o s 2 c + cos2 6 cos 2 c — sin*6 sin 2 c s i n M T h u s w e find n o w : sin 2 £ sin 2 c s i i i M = 1 — cos a a — cos 2 6 — cos *c + 2 c o s a c o s b c o s e T h i s is positive. W e define n o w a positive X so t h a t : A r 2 sin , a s i n ' è sin 2 c =

1 — cos2a — cos'ò — cos'c + 2 cos

sinM , Thus: — — 2 = 1 , X* san a m

sinM ,„ - —2 - = X*, sin a

a cos b cos c

sin Λ + -— = X sin a

T h e r e is o n l y t h e positive sign b e c a u s e A a n d a a r e b o t h < 180°. W e find i n this w a y t h e sine rule. sin A sin Β — = -—sin a sin b

sin C — , or written out: sin c sin A sin b = sin Β sin a sin A s i n e = sin C sin a

sin Β sin c = sin C sin b

\ >

(3)

J

F o r C = 90° t h e s e c o n d e q u a t i o n o f t h e a b o v e n o w gives : • sin AΛ =

S

Ì

n

a

/ A \ (4)

sine (3) W e s t a r t w i t h t h e s e c o n d expression of t h e c o s i n e r u l e a n d u s e this r u l e o n c e m o r e . sin a sin c cos Β = cos b — cos a cos c = c o s b — (cos b cos c + sin b sin c cos A) c o s = cos b — cos b cos 2 c — sin b sin c cos c cos A = cos b sin 2 c — sin b sin c c o s c cos A

c

20

Properties of Double Stars

Division by sin c gives the third rule : sin a cos Β = cos b sin c — sin b cos c cos Α Λ sin b cos C = cos c sin a — sin c cos a cos Β > sin c cos A = cos α sin è — sin a cos b cos C J

(5)

For C = 90° the last line gives : sin c cos A = cos a sin b The cosine rule gave : cos c = cos a cos b Division gives : tan c cos A = tan b Thus : cos A =

(6) tane There is also another proof of the third rule. In Figure 2 express χ in Δ DBC and Δ DAC with the cosine rule. o

cos χ = cos a cos 90 + sin a sin 90 cos Β = cos b cos (90—c) + sin b sin (90— c) cos (180-Λ) sin a cos Β = cos b sin c— sin b cos c cos A

Astrometry

21

2. Trigonometric parallax. Stellar parallax means the maximum difference in the lines of sight to a star as seen from the earth and the sun. Let us assume first a circular revolution of the earth around the sun in the plane of the ecliptic. Because the earth's orbit is an ellipse, we thus introduce a very small error, but we will correct for this later. The parallax is also the maximum angle an observer on the star "sees" the distance sun-earth, or the angle the astronomical unit subtends at the star (Figure 3).

Figure 3. The circular revolution of the earth E in the plane of the ecliptic causes the star to describe a parallactic ellipse in the plane of the sky.

22

Properties of Double Stars

The star as observed from the earth has an apparent movement about the position as observed from the sun which is called the heliocentric position. In space in the direction of the star this will be the exact reflection of the earth's orbit and is thus a circle in a plane parallel to the ecliptic plane. Observed in the plane of the sky we see it thus projected as an ellipse with the semi-major axis equal to the radius of the circle. This semimajor axis is called the parallax. In the line of sight we have also a yearly periodic motion by which the radial velocities are affected. This is independent of the distance and the results are measured directly in kilometers per second. The astronomical unit can be determined from these radial velocity measurements as we will see later. Let 7Ta be the absolute parallax. If the parallax is expressed in degrees and the distance r in astronomical units we have according to the definition of parallax (Figure 4) : S

heliocentr

E Figure 4. Parallax is the maximum difference in the lines of sights as seen from the earth and the sun.

Astrometry

23

sin πΛ = (7) r Because π, is very small we can omit the sine if π, is expressed in radians. If we express wa in seconds of arc we have : 206,265 ιτΛ = r We often use the parsec as a unit of distance. One parsec is the distance at which a star has a parallax of one second of arc. It thus equals 206,265 astronomical units. If πα is expressed in seconds of arc and r in parsecs the relation becomes simply : Ta = J

(8)

In Δ S®E we have according to the sine rule : sin (θ - Θ') = - sin θ r Again (θ — Θ') is small. Substitution of (8) gives a relation which is independent of the units used. (θ - Θ') = 7Ta sin θ (9) As seen from the earth the heliocentric direction, the geocentric direction, and the direction towards the sun are in one and the same plane through the earth, sun and star. The geocentric position always will lie on a great circle between the heliocentric position and the sun. 3. Parallax in celestial longitude and latitude. Let S (λ, β) be the heliocentric position, S' (λ', β') the geocentric position, and the sun (®, 0) the sun's position in Figure 5. Let P^ be the pole of the ecliptic. SS' ® is a great circle and SS' = θ - θ'. Further SU is made parallel to the ecliptic. If (14) - COS a sin δ COS ® } J If we work out our result for the longitude and latitude we can see that the underlined terms are introduced by the rotation.

28

Properties of Double Stars

The e is a constant for a long time interval. The α, δ are constants for the star after correction for precession. The longitude of the sun ® varies with time and has a period of a year. For a certain date this is also a constant and we find in this way the coordinates of the unit parallactic ellipse = 1. 6. Parallax factors. These are the coordinates of the unit parallactic ellipse for a certain date. In case of longitude and latitude we have: P\ = sin ( ® - λ), Ρ β = - cos (® - λ) sin β (15) The longitude of the sun can be found in the almanac for each date. The values of Α, β can be computed from the known values of α, δ with help of the parallactic triangle Ρ„,Ρ«£ in Figure 8

Figure 8. The parallactic triangle P^P^S. sin β = cos e sin δ - sin e cos δ sin α = cos SP« cos A cos ß= COS δ c o s a = cos S t > sin A cos β = sin e sin δ + cos e cos δ sin a = cos SQ J

The three expressions are nothing more than the direction cosines of the star with respect to the ecliptic.

29

Astrometry

In case of right ascension and declination we proceed in the following way. For simplicity define: q = -sina χ Ρ = COS e COS O, b = — cos a sin δ J a = sin « cos δ — cos « sin α sin δ, The expressions for the parallax factors are then : Pa = ρ sin ® + q cos ® X Pe = α sin ® + b cos Θ X

S

(17)

The p, q, α, b are constants for the star under observation. The coordinates P„ Pt are thus linear functions of sin ® and cos ® and depend consequently only on the date. The angle ν in Figure 9 is situated between + 23¿° and - 2 3 f o r the given star. For right ascension 6h and 18h this angle is zero because the ecliptic and equator run parallel here. north

8 β geocentric λ cos β α COS S

eos t

*

~ ~Γ 1 /•1 - // / ~" r ^ t ^ Ρ Pa IPb N. \ O iT-r^ J' h e l i o c e n t r i c

\

P*

\

1

Figure 9. Parallax factors are the coordinates of the unit ellipse. In our derivation we assumed a circular orbit for the earth with a radius of one astronomical unit. This is not exactly true because we know the true orbit to be an ellipse. The radius vector of the earth's orbit or the distance to the sun R is very close to unity and is given in the almanacs. The largest difference

30

Properties of Double Stars

is 0.017 astronomical unit. If we take this into account, we have always instead of πΛ in our formulae. It is customary to include the factor R in the final stage in the expression for the parallax factors, which should be computed to three decimals. One can note here that the parallax will be determined best when Ρ λ or P a are near + 1 ornear — l,thus for® — λ = 90° or 270°, thus ® = λ + 90° or ® = A + 270° = λ - 90°. For most accurate results one therefore has to observe the parallax star when the sun differs 90° in longitude from the star. For the star in the meridian this happens at the beginning and end of the night. 7. Proper motion and parallax. Let nt be the relative parallax with respect to the comparison stars. A single star has a linear proper motion μ and a parallactic ellipse with semi-major axis 77r (Figure 10). The heliocentric path of a star is given by cx + μχί, Cy + μγί, linear with the time t in years. For the geocentric path we have to add π,Ρα, nTPt. The result is an equidistant spiral. Except for some circumpolar stars we can observe only half of this spiral because the angular distance between the sun and star is too close for the other half of the year. ξ =

Cx +

μχί + πτΡα

i

η =

Cy +

μ,ί

)

+ τττΡa

^jg-j

The known factors are the measured quantities ξ, η, the time t and the parallax factors Pa, Pa. Unknown are cx, cy, μχ, μ,, πτ and they have to be determined from the observations. The constants cx, Cy are added because the proper motion will not go exactly through the origin. A graphical method to determine the constants roughly is as follows. Take two observations which are exactly one year apart. They give us the proper motion and its components μ„ μ^ thus we have now size and direction of the proper motion. Now

Astrometry

31

determine the width of the strip in which all observations are situated. We know also the unit ellipse in its true orientation. Decrease this ellipse in such a way that it fits in the strip. The semi-major axis is the parallax -π,. The straight line of the proper motion gives for / = 0 the constants c„ cy. In practice we compute this still more accurately. Because the parallax is so small, we have to take many precautions in taking the observations, in the measurement, and in the reduction in order to get reliable results, and these we must study first.

α cos δ / /

t

'

* h'''

/

Figure 10. The observed equidistant spiral for a single star consists of proper motion and parallactic ellipse.

32

Properties of Double Stars

8. Photographic stellar parallaxes. At present parallaxes of about 6000 stars have been determined. F. Schlesinger was the first to study the basic precautions one has to take in photographic parallax work. One needs a refracting telescope of long focal length because then the scale is large. The scale factor is defined as the number of seconds of arc per mm. It can be found by measuring the distance between two known stars in an open cluster for which the right ascensions differ but the declinations are the same. It is best to take this plate during meridian passage. For accurate work one has to take the refraction into account which in the vertical circle is given by the refraction constant times tan z, where ζ is the zenith distance and the so-called refraction constant is the atmospheric refraction at ζ = 45°. to o b j e c t i v e

t positive lens

o sharp f o c u s

o 20 focal setting

plate sensitivity

Figure 11. The focal curve of an objective.

c S? Sw c S! — υ O 60 Λ ex ω et | Eυ 2 ίΛ ~ cd "2 cd ·· .22 o CO : -o OD ω J O cd X) gO eω ex o £e cd cd Τ3 4) υ tri cd u. Ji .22 u CU ό -a « 2I u e 03

£

>

SCl,M Ci g>CU ^ o ε " ιOSQ o il ε cd o < 2 υ-> E U ¡3 r\. cd §·£ § O ex. « "§ « 1 » χ" U , S w ow . crt O cd S jS P-. (*3 - x't) x

i

We usually use here the symbol [ ] instead of the Σ sign. χ = [Dx]

+ X' -

Y=

+ r

[Dy]

[Dx']

(25)

- [Df]

The [Dx] is found on the standard plate, and the [Dx'] from the other plate. We can thus correct X' to X in our standard system. 12. Method of dependences. For more than three comparison stars we cannot give such a geometrical picture. The number of equations is no longer equal to the dependences to be found. We have to proceed by least squares. For simplification we take the zero point in such a way that the average value of the χ and the y are zero for the η comparison stars on the standard plate. Thus [x] = [j] = 0. This determines the gravity center G of the

*iyi

±9 XG

lieq,

tQ--

1950

0

Il e q , t i m e

t

*îV} Figure 16. Standard plate and reference frame. G is the gravity center. H is the dependence center.

41

Astrometry

triangle in Figure 16. We will solve only for the x-coordinate ; the j-coordinate goes in the same way. Our linear equation of condition was: axx

+

b

x

y

+

cx

OyX

+

b

y

y

+

C y = y

=

χ

-

x' - y '

For the x-coordinate the normal equations are : [χ2] ax F [xy] bx = [x(x-x')] [ x y ] a

x

+

[ f ] b

x

ncx

=

[ y ( x - x ' ) ]

=

[(x—x')] = — [ * ' ] J

The unknowns ax and bx can be solved with help of determinants. [x(x-x')] [xy] IXx-x')] [y*] ax -etc. -

[ χ · ]

[xy]

[xy]

[ / ]

The [χΐ]> [xy], [.y*] are constants from measurements on the standard plate. Therefore: [y*] [x(x -x')] - [ x y ] [>] = 1 because [χ] = [ j ] = 0. Also [Z>2] = minimum because of the least squares procedure. Notice that only standard plate data enter in the formula. The dependence reduction can now be written as : X=X'

+ [D{x-x')]

= [Dx] + X' - [Dx'] = [.Dx] +ξ

Υ = Γ

+ [D(y-y')ì

= [ D y ] + Y ' - [Dy'] = [Dy]

+v

(28)

The dependence center H in Figures 16 and 17 has as coordinates [Dx], [Dy]. With respect to this center the coordinates become for the observed point on our spiral : ξ = X' - [Dx'] = X' - (D1x\ + D*x't + Dtx'z + ...)! •n~Y'~ [Dy'] = Y' - ( D ^ + D,y\ + Day'a + ...))

ν

U-

lold

. K

)

Y'

7Γ -4Y' fi

X' *¿y i

xiyi Figure 17. Provisional reference frame of another plate. The parallax star moved from the old position to the new position π.

44

Properties of Double Stars

14. Computation of dependences. We use the measurements of the comparison stars and parallax star on the standard plate. Call: Xj [y2]

f = Jl

M*2]

- y> [xy] S

[χ»] [ f ] - [xy]* '

'

-

*

[*«] [ / ] -

M [xy]*

1

;

These depend on the coordinates of the comparison stars only. We see that [fx] = [gy] = 1. The dependences can now be written in the following linear relation : D , = f

i

X

0

+ g , Y

0

+ η

(31)

For the three comparison stars in a triangle with equal sides and the parallax star in the gravity center we find A = 1/3. The f¡, gi tell us about the asymmetry ; the X0, Y0, describe the deviation of the parallax star from the gravity center. Because M = [j] = 0, it must follow strictly: [£)] = 1. For the standard plate we tried to make ξ = 0, η = 0, so that X = X0, Y = Y0 and within the rounding off errors it follows [Z)x] = X0y and [Dy]

=

Y0.

We can also compute the annual variation AD by differentiating: ΔΑ=/μχ (32) with the controls : [ΔΖ>] = 0, [Δ Dx] = μχ, [ΔΖ)^] = For small proper motions of the parallax star it is not necessary to change the D's, but for large proper motion two or three sets of D's may be advisable. We compute the dependences to three decimals using only the standard plate. For the other plates we then find the position ξ, η with those dependences on the correct coordinate system. 15. Number of had the form:

comparison

ξ = X' - DlX\

stars.

The solution for every plate

- D¿c'g - D3X'3

- ... -

Dnx'n

45

Astrometry

The square of the probable error of this solution is: ζ=

e

V '

Λ

+

A

-

Χι+

D

*

'

Χ 2+

D

»

e

Χ s+

* .

···

+

D

*

6

2

-

Assume that the probable errors of the measurements of all stars are the same namely e ; it follows : = «»(1 +[D·]), e = e V(1 +[/>«]) ξ ζ If the stars are well divided over the plate, so that β = g¡ = 0 we have D¡ = 1/m, and we get then : « = e V ( l + -s) = * V " ( l + - ) (33) ξ η* η For three equal dependences the square root is 1.15; for four we find 1.12, etc. Usually one takes three or four comparison stars and measures the parallax star twice. More comparison stars take too much measuring time for the slight increase in accuracy. How does the distribution of the comparison stars over the plate influence the probable error of the result? We can compute this for different cases. It can be seen then that the distribution and thus the dependences may be quite unequal, and that the parallax star does not have to be situated in the gravity center. Only when the parallax star lies outside the configuration do we get negative dependences which we have to avoid. 16. Relative and absolute trigonometric parallax. Usually one takes two glass plates per night. Taking more is useless because of systematic night errors. There are also plate errors and measurement errors. The order of these probable errors are respectively: ± 0T012, ± 0T023, ± 0T015,combined ±0.030. All measurements for one night are averaged to a mean night value.

46

Properties of Double Stars

A least squares solution is made for which the equation of condition is according to formula (18) : ξ = Cx + μ*ί + nTPA

\

η = Cy + (V + πτΡβ

j

One solves these equations for each coordinate first separately and then combined. The relative parallax πτ, the proper motion (μ*, (¿y) and the zero point (c„ cy) are found in this way. Also the probable errors follow from the computation (Plate II). We measured the relative ellipse of the parallax star with respect to the comparison stars. Those comparison stars are far away but not at an infinite distance. In other words they have a small absolute parallax, which has to be added to the relative parallax of the π star to reduce this value to an absolute one, τra. Figure 18 shows why one has to add this correction. It is still better to add the average parallax value of the comparison stars or [DttC], The parallaxes of the comparison stars are found in a statistical way from proper motions and are given as a function of apparent magnitude and galactic latitude. The most recent values are given by A. N. Vyssotsky and E. T. R. Williams derived from secular parallaxes, and by L. Binnendijk from peculiar motions according to a method of J. H. Oort. The probable error of a good parallax determination is of the order of ± 0"005. However, it was found that there are systematic differences between results of different observatories for the same star. The amount may be 0.003 and can be explained if the number of evening and morning plates are not the same. There is also a periodic error for which one can apply an empirical correction. Very probably the result is caused by temperature changes in the objectives. Some stars are observed on winter mornings and summer evenings. The reduction curves of different observatories are not the same, which is surprising. These corrections have been applied in the Yale Catalogue of

Astromeiry

47

Parallaxes to produce a uniform system of absolute parallaxes. This catalogue is very important because all other distant measurements are indirect and have to be calibrated with the trigonometric parallaxes. Thus the trigonometric parallaxes form the basis of all distances measured in the universe. To know those distances in kilometers or miles, the astronomical unit has to be known in kilometers or miles. There are several ways to find this. The orbital motion of the minor planet Eros provides an excellent opportunity to determine this distance. The daily parallax of the sun was found to be 8T80, corresponding to a distance of 149.5 million kilometers.

r e l o f i ve ellipse absolute ellipse

E Figure 18. The relative and absolute ellipses.

48

Properties of Double Stars

17. Proper motions. The proper motions of many stars have been measured and published in proper motion catalogues. An example is the Radcliffe photographic catalogue. One takes plates at least 20 years apart in time at about the same date of the year. Measures are made relative to the faintest stars on the plate. A correction of the form (ax + by 4 c) must be applied. The probable error in one coordinate in the Radcliffe catalogue is ± 0-003; in case of the Pleiades catalogue of proper motions by E. Hertzsprung the probable error is ± 0*001 in each coordinate. One next tries to derive absolute proper motions; and a first sequence of plates has been taken at Lick Observatory showing the extra-galactic nebulae. A difficulty which still remains is that the images of stars and nebulae do not look alike. Accurate proper motions are important for a number of problems. The peculiar motion of a certain star is usually unknown, but for a large number of stars the sum will be zero if the velocity distribution is at random. The problems are thus statistical in nature, and one can find the precession constant, the linear solar motion, the galactic rotation effect, mean parallaxes, etc. A complication appears because the real distribution of stellar velocities shows a preferential motion, which can be explained by the theory of the galactic rotation. 18. Solar motion and apex. The point toward which the sun is moving in space with respect to the center of the nearby stars is called the apex of the linear solar motion. It is in the constellation of Herculus. It can be found from proper motions and radial velocities of stars. The stars situated near the apex seem to move towards the sun and will show a negative radial velocity on the average while near the antapex they will be receding and show a positive radial velocity. Exactly 90° away the effect in the radial velocity is zero, while that for the proper motions

Astrometry

49

is at a maximum, opposite in direction to the solar motion and depending in size on the mean distance of the star group. In Figure 19 let the sun be at the origin and the rectangular coordinates of a star x, y, ζ or in polar coordinates r, a, δ. We have then the relations : X = r cos δ cos α 1 y = r cos δ sin α I (34) ζ = r sin δ J By differentiation we get : dx dr . d8 da — = — cos δ cos α — r sin δ cos a — — r cos δ sin α — dt dt dt dt dy dr . db da -r- = — COS δ sin α — r Sin δ Sin α — + r cos δ COS α — dt dt dt dt dz dr . dh — = sin δ + reos δ — dt dt dt

Figure 19. Rectangular and polar coordinates of the star S with respect to the sun at the origin.

Properties of Double Stars

50

We will call the velocity components u, v, w. Further the radial velocity VT = drjdt and the proper motion components μα = da/dt and μ^ = dB/dt are observed. We have then in the new notation: U — Vt COS δ COS α — Γ μΛ sin δ COS α — Γ μ 0 COS δ sin α V = Vr COS δ sin α — r μβ sin δ sin α + r μ α COS δ cos α (35) W = Vt sin δ -ι- ΓμΛ COS δ Solving for Vt, μ,», μα cos δ gives : VT μΛ

= =

μ α COS δ =

+ u cos α cos δ + ν sin α cos δ + w sin δ U r u

.

.

V .

COS α Sin δ S· i n a

r !

v

.

^

W

S i n α S i n δ + - COS δ

(36) ·\

r

COS a

J For a single star we do not know the peculiar space motion in advance. However, we can assume random motions of the stars. For all stars in one magnitude interval the peculiar radial velocities will cancel in the mean value. The same holds for the peculiar proper motions if we assume that all stars in the magnitude interval have the same distance to the sun. We will consider therefore further on the mean values of the stars concerned. The velocity components u, v, w are then the components of the reflected solar motion. From the radial velocity observations we see that we can derive the u, v, w by least squares, but from the proper motion observations we find only the w/r, v/r, T

T

wjr.

Let V@ be the linear solar speed directed towards the apex with coordinates A, D. The speed - V@ will have then the components u, ν, w, as observed from the motions of the stars. From a similar figure we find a similar relation as formula (34). u = - V& cos D cos Α Λ v=

- V@ cosDsinA

w=

- V@ sin D

I

J

(38)

Astrometry

Further we have: u2 + V2 + >vs = V*

51

(39)

u ujr (40) sin D w wjr wjr tan D cos D v'("2 + V2) V{(u¡rY + (v/r)*} From the radial velocity observations both the size V@ and direction A, D of the apex can be found ; from the proper motion observations only the direction or the position of the apex in the sky can be derived, as could be expected a priori. It can be seen from the last formulae that the units used are of no importance as long as they are consistent. The results found from the nearby stars are V@ = 20 km/sec, A = 18\ D = + 30°. It should be noted here that we find the speed and direction of the relative solar motion with respect to the nearby stars. If we study more distant objects we find another relative speed and direction. Finally when we study the globular clusters we find the motion of our sun with respect to the center of the Milky Way system. 19. Secular parallax. We have seen that the sun moves through space towards the apex with a speed of about 20 kilometers per second with respect to the local standard of rest. The reflected space motion of a star is the same 20 kilometers per second directed towards the antapex. If the angular distance between antapex and star is called σ the radial velocity of the star will now be 20 cos σ kilometers per second as far as the reflection of the solar motion is concerned (Figure 20). We have: one astronomical unit/year = 149.5 million km/ 31.56 million seconds = 4.74 km/sec. The sun moves 20/4.74 or 4.2 astronomical units through space per year relative to the nearby stars. In the tangential plane this reflected speed is 20 sin a kilometers per second or 4.2 sin σ astronomical units per

52

Properties of Double

Stars

year. The proper motion component in the antapex direction depends in addition on the distance of the star. If the star had no peculiar motion, then this shift in the plane of the sky towards the antapex would determine the distance of the star. Unfortunately the peculiar motion is unknown for a single star but for a group of similar stars one can again assume that they cancel in the mean value assuming random motions of the stars. This gives us the secular parallax which is thus a statistical parallax. 20 c o s σ

— 2 0

opex

km/sec

antapex

Figure 20. The secular parallax is 4.2 times as large as the trigonometric or yearly parallax. The base line is 4.2 times as long per year as the distance sunearth and moreover this base line increases throughout the years. If πΜ means secular parallax, and πΛ the absolute trigonometric or annual parallax, we have accordingly the following relation: = 4.2 π,a (41)

Astrometry

53

Call the component of the star's proper motion in the antapex direction ν and perpendicular to it τ (Figure 21). Then Στ = 0. As before independent of the units used it follows : υ = 7τκ sin σ

- 4 . 2 77a

sin σ,

(42)

Sina

antapex

a cos 8

Figure 21. The observed proper motion μ of a star with the components υ, τ. The peculiar proper motion of the star may be Upe. We can measure the υ while we can compute the σ. However, the weights are not the same for all stars, but are largest 90° off from the apex direction. There one sees the largest deflection of the solar motion in the proper motion of the star. If e is the probable error then we measure ν ± e thus weight ρ = 1/e2. υ

« , sin*a . . ± —» P = ~r=P sin'a (43) sino sino If for all stars the measured proper motions are of equal accuracy, «ι = ea = . . . andp' = sin2 σ. Therefore : "l sin2 σχ + . . . sin®. sin2 »! + .

[υ sin σ]

ν sin σ

[sin2 σ]

sin2 σ

(44)

However, when the accuracy is not the same, which is more probable, we have Φ