sin 2(H + a) sec 2 <5 - CHAPTER 6 Aberration

= -JÄ;² sin 1" {(1 + cos²*) sin 2a cos 2 0

—2 cos e cos 2a sin 2 0 } sec²d + \k² sin 1" sin²€ sin 2a sec²<5,

in which the constant term must be distinguished from elliptic aberration.

Similarly δ' — δ contains the constant part

— \k² sin 1" {(1 + cos²e) — sin²e cos 2a} tan δ.

These are automatically included in a second-order reduction to apparent place, unlike the elliptic aberration, which is explicitly removed from the reduction and included in the geometric place.

To show the order of magnitude of the effects added to (74), the component of the annual aberration due to the barycentric motion of the Sun alone, or to the action of any particular planet, may be obtained separately by sub-stituting the velocity components of only this motion in Eqs. (64) and (65).

With sufficient accuracy for this purpose, the center of mass of the Sun and a planet of mass m may be taken to be at distance ma from the Sun, where a is the mean distance of the planet, and the Sun considered to move around this point in a circle in the orbital plane of the planet, with an angular speed equal to the mean motion n of the planet. Neglecting the latitude of the planet, the rectangular coordinates of the Sun referred to this point as origin

STELLAR ABERRATION 123 and to axes parallel to the equatorial system, are

mtfcos(180° + /), ma cos € sin(180° + /), ma sin esin(180° + /),

where / is the heliocentric longitude of the planet; hence the velocity com-ponents are

+ nma sin /

— nma cos e cos /,

— nma sin e cos /, and therefore

/ t \n (n)"ma ( . . , , n ©

(a — a) = — — {sin a sin / + cos c cos a cos /} sec o, c

(ό' — δ)" = — — {cos a sin <5 sin / + cos I (sin € cos δ c

— cose sin a sin ό)}, in which the constant coefficient may be expressed as

(nYma f, n

— = kma —

c n@

where T is the sidereal period of the planet in sidereal years.

The values of the coefficient k'majT for the effects of the different major planets are:

Venus (T.0001 Saturn 0".0019 Earth 0".0001 Uranus (T.0002 Jupiter 0".0086 Neptune 0".0002

The stellar aberration produced by the motion of the Sun relative to the center of mass of the solar system is therefore virtually all due to the actions of Jupiter and Saturn; the corrections required to the velocities X\ Υ', Ζ' per half day to represent the effect of Jupiter are

-0.00000 36 sin lj9 +0.00000 33 cos lj9 +0.00000 14 cos lj9

and for Saturn

-0.000 0008 sin ls, +0.000 0007 cos ls, +0.000 0003 cos ls. The stellar aberration due to the motion of the Earth around the center of mass of the Earth-Moon system may likewise be separately obtained from the equatorial rectangular coordinates of the Earth relative to the center of mass, which are, with sufficient accuracy

Ax = —/cos j8j cos Aj

= —/cos ôj cos a5,

Ay = —/cos /Jj (cos € sin A^ — sin e tan /?))

= —/cos oj) sin aj,

Az = —/cos jSj (sin € sin A^ + cos e tan β})

= - / s i n ôp

in which

/ =^m}lm@ 8".80 sin 1" cosec π, 1 + mj)/m@

= 0.0000312.

Neglecting the latitude of the Moon, the velocities of the Earth relative to the center of mass may be taken to be

+/sin A]) —- = +fn'i sin 1" sin % dt

—/cos € cos A}) — = —fn'i sin 1 "cos e cos J), dt

—/sin € cos A}) —- = —fn'i sin l^r/ sin € cos % dt

where np }) are the mean motion and mean longitude of the Moon. Then by Eqs. (64) and (65), with

__ I = _ 0.0000312 x 1.3176 x 3600 c "" 186300 X 8640

93000000

= -o^ooese,

STELLAR ABERRATION 125 we have

cos ô (α' — α)" = — 0".0086{sin }) sin α + cos € cos }) cos α}, (ό' - δ)" = -0".0086{sin D cos α sin ô

— cos € cos J) sin α sin ô + sin e cos ]) cos <5};

i.e., the lunar aberration may be included in (71) by adding — 0".0079 cos }) to C, and -0".0086 sin }) to D.

The coefficient — 0".0086 is the mean displacement in the plane of the lunar orbit and is therefore equal to the product of k' by the ratio of the velocity of the Earth around the Moon to the velocity of the Earth around the Sun,

— k'ain^Kl + m®jmj) _ , ,f m

a@ⁿ®ⁿ®

in which n^jn@ = 13.37.

The numerical coefficients of the lunar aberration in. right ascension and declination may be obtained somewhat more accurately from the rate of change of the principal term (6".454 sin D) of the lunar equation than from nr In the plane of the ecliptic, the mean aberrational displacement is

V — - (6".454 sin D) = 6".454/c' cos D ^ ^

n@dt n@

= 20".47 x 6".454 x 12.37 sin 1" cos D

= 0".0079 cos D,

with a period of 29^d.5. Neglecting the latitude of the Moon, therefore, the lunar aberration is obtained by adding — 0".0072cos ([ to C, and

—0".0079 sin i to the day number D .

For determining the aberration in right ascension and declination near the pole, where the declination becomes so large that a calculation to greater accuracy than the second order may be needed, the rigorous trigonometric reductions Eqs. (59) and (60) may be adapted to practical use by expressing them in terms of the day numbers. Equation (59) becomes

, . sec ô{C cos a + D sin a}

tan(aapp — a) =

1 — sec ô{D cos a — C sin a}

h sin(H + a) sec ô

1 — h cos(H + a) sec ô ' (79)

and putting

k sin K = C tan e,

k cos Ä: = h cos {// + a + 4(aapp - a)} sec £(aapp - a), Eq. (60) becomes

tan(oapP - 8) = ' . (80)

1 — k cos(K + ô)

The second-order expressions (77) and (78) may be obtained directly by developing these rigorous expressions.

Stellar Aberration in Rectangular Coordinates

From the rigorous equations (57), in terms of the direction cosines / = cos a cos <5,

m = sin a cos ô, n = sin δ, with the slight approximations

cr = c, z = y tan e,

and with the velocities corrected to remove elliptic aberration by putting x/c = — D, y\c = + C, we obtain

/ ' - / =

- A

rri

m

= +C,

ri — n = + C tan e.

Differential Stellar Aberration

Since the value of the stellar aberration depends upon position on the celestial sphere, the aberration in general alters the positions of the celestial bodies relative to one another. For example, the apparent angular distance and position angle of one object relative to another differ from the values for the geometric positions of the objects.

In terms of equatorial coordinates, denoting the difference of the geometric positions by Δα = α2 — ο^ and Δό, and the difference of the apparent positions by Δα' = α^ — aj[ and Δ<5', to the first order

A / A <K^a' — a) A 3(a' — a) A

Δα = Δα + -¹— ^Δα + -^— -Δί, 9α σδ

STELLAR ABERRATION 127 and similarly for Δό'; from (71) and the expressions for the star constants, therefore, the corrections to be added to the geometric differences in order to obtain the apparent values are

Δ(α' - a)^s = Ckc + DM

= + h(D cos a — C sin a) sec δ Δα⁸ sin I^s + £5(D sin a + C cos a) sec δ tan Ô Δό" sin 1"

= + hh sec à cos(H + α) Δα⁸ sin l⁹ + hh sec à tan δ sin(H + α) Δδ" sin 1", Δ(<5' ~ δ)" = - ( £ sin a + C cos a) sin δ Δα⁸ sin I^s

+ (D cos a — C sin a) cos δ Δδ" sin 1"

- Ctanesin(5A<5"sinl"

= —A sin 5 sin(H + α) Δα⁸ sin I^s + h cos δ cos(H + a) Δό" sin 1"

- i sin ί Δί^# sin Γ.

Subtracting these corrections from the observed values of Δα', Δό', for an object relative to a comparison star, measured in the sense of "object minus star," reduces them to the geometric values, from which the position of the object may be obtained from the catalog place of the comparison star;

tables for facilitating this reduction are given in the national ephemerides.

The maximum values of the corrections for two objects 1° apart is of the order of 0⁸.02 sec δ in right ascension and 0".3 in declination.

In the ecliptic system, to the order of accuracy of the equations (75), Δλ' = Δλ + ( f l ' ~ f l ΔΑ + (λ' - λ) tan β Δβ,

sin ß cos ß

Δ0' = aß + (β' - β) Qoißtiß - (λ' - λ) sin ß cos βΔλ.

Stellar Aberration in Apparent Distance and Position Angle

The differential aberration in angular distance s and position angle p may be obtained from the first-order expressions ssinp = Aacoso and s cos/? = Δό. Differentiating these expressions,

cosp Δ? — s sinp Δρ = Δ(<5' — δ),

sin/? Δ^ + s cosp Δ/? = Δ(α' — a) cos δ — (ό' — δ) sin δ Δα.

Substituting the expressions for Δ(ό' — ô) and Δ(α' — a), and for δ' — ô9

Δα, and Δό on the right, and solving for as and Δ/?,

s' — s = s{D cos ô cos a — C(tan € sin ô + cos ô sin a)}, p' — p = C tan ô cos^OL + D tan ô sin a;

or in terms of the day numbers H9 A, /,

As = s{h cos(H + a) cos ô — / sin ô}, Δ/7 = h s\n(H + a) tan ô.

All distances from the same point, whatever their directions, are therefore changed in the same ratio, which at maximum is less than 5/10,000; and all position angles from this point are changed by equal amounts.

Diurnal Aberration

The linear velocity in the diurnal rotation is directed eastward, perpendicular to the plane of the meridian ; the aberrational displacement on the celestial sphere is therefore along the great circle arc Θ from the geometric position to the east point of the horizon. In terms of the geocentric coordinates p and φ of the observer, and the local sidereal time θ, the radius of the diurnal circuit is p cos ψ' and the linear speed is v = (p cos φ')(αθΙώ); denoting {vjc) cosec 1" by /c, the amount of the displacement in seconds of arc is therefore κ sin Θ toward the east point.

It is immediately evident from the geometric relations on the celestial sphere that, to the first order in v/c, the diurnal aberration in altitude H, and in azimuth A reckoned from the south point positive toward the east, may be obtained from (75) by replacing k' by κ, β by H, λ by A, and 0 by 180°:

^aPp — A = +K sec H cos A,

#apP ~ H = —K sin H sin A.

Likewise, the diurnal aberration in hour angle h and declination ô is obtained by replacing k' by #c, β by ô, λ by 360° - h, and 0 by 180°:

^ a p p "~^{n =} ~~^{K S e C} ^^{C 0 S}ⁿ->

Oapp — ô = +K sin δ sin h.

The aberrational displacement in right ascension is the negative of the displacement in hour angle.

The diurnal aberration in right ascension and declination may also be obtained from Eqs. (64) and (65). The geocentric rectangular equatorial

STELLAR ABERRATION 129 coordinates of the observer, with the A'-axis directed toward the vernal equinox, are

x = p cos φ' cos 0, y = p cos ψ sin 0, z = p sin <p\

whence the linear velocities in the diurnal motion are

dxjdt = — p cos 99' sin θ(αθ/ώ)9 dyjdt = + p cos 9/ cos 0(</0/Λ)>

and therefore, to the first order in v/c,

1 Af\

(α' — a)" = p cos 99' cos(0 — a) sec ô cosec 1", (ό' — ôy = p cos 9/ sin(0 — a) sin ô cosec 1", in which

p cos φ = — = = - cos 99, VI — ^ sin²9?

where φ is geodetic latitude, a0 is the equatorial radius of the Earth, and e0

the eccentricity of a meridian section. The factor 1/%/1 —el sin²<p varies only from unity at the equator to 1.0034 at the poles.

The factor (a0/c) ddjdt is known as the constant of diurnal aberration.

The period of Θ is one sidereal day, or 0.99726 95664 mean solar days, hence the ratio of ddjdt to the mean orbital motion n of the Earth (which has a period of one sidereal year) is 365.256360/0.997269566 = 366.2564 (the length of the sidereal year in sidereal days); and in terms of the constant k' of annual aberration, the constant of diurnal aberration is

^ — = 366.2564 sin π& (1 - e²)^1/2k' c dt

= 366.2564 x 8".80 sin 1" x 0.999860 x 20".47 sin 1"

= 0".3198 sin 1"

= 0^s.02132sin Γ.

This constant is the ratio of the linear velocity to the speed of light; and with the day as the unit of time, it is therefore also given by

2πα0 ,, cosec 1 .

86,400 a0

498^s.58 8".80 sin 1"

Therefore

/ / \// 0 .3198 /Λ \ s

(a — a) = cos w cos(0 — a) sec <5, V 1 — ej sin ç>

0" 3198

(δ' — à)" = cos φ sin(0 — a) sin δ.

V I — ^ο sin² 9?

On the meridian, where 0 = a, the diurnal aberration in declination vanishes, while the diurnal aberration in right ascension attains its maximum value (0".3198/v 1 — e², sin²<p) cos φ sec <5 which in the zenith, where δ = φ9

reduces simply to the constant.

Eliminating (0 — a) gives

(cosoAa)² (Δό)² = j k² cos²9? k² cos²ç> sin²<5

the equation of an ellipse with semimajor axis k cos φ along a parallel of declination, and semiminor axis k cos φ sin <5 along a circle of declination.

Planetary Aberration

Correcting the apparent position at time / for the stellar aberration due to the motion of the observer at the instant t gives the position at the time t of the place where the object was geometrically located at the previous time t — Δ/ when the light left it. The further correction for the motion of the object during the light time Δ/, which is required for determining the position at a given time of the place where the object is actually located at this same time, may be made in either of two ways. The geometric position at t — Δί relative to the geometric position which the observer had at that instant may be determined by applying a correction — Δ/ to the time of observation, leaving the observed position unchanged; or, the geometric position at time / relative to the position of the observer at the instant of observation may be found by correcting the observed position for the relative motion of the Earth and the object. Both of these procedures automatically include corrections for the motions of both the Earth and the object; the stellar aberration does not explicitly appear, but is implicitly included in the planetary aberration.

Because of the simplicity of merely antedating the time of observation, this procedure is widely followed; however, the method of correcting the observed place is perhaps preferable, since it gives the geometric position at the instant of observation, and it is theoretically preferable because it brings out the important principle that aberration depends upon the relative

PLANETARY ABERRATION 131 velocity of observer and source. Both procedures are independent of the motion of the system as a whole.

When the motions are rectilinear and uniform, so that the displacements in space are proportional to the velocities, the planetary aberration may be obtained from the velocity of the Earth relative to the planet by means of a geometric construction exactly similar to the one by which stellar aberration is obtained from the total velocity of the Earth (see Fig. 24). The velocity of the body may be expressed in the form

VP = V@ + (Kp - V@).

During the light time, the component V@ displaces the body geometrically from its initial position P by the same amount, and in the same direction, as

FIG. 24. Planetary aberration; cf. Fig. 19.

this position P at time t — A n s apparently displaced by the stellar aberration produced by the motion of the Earth; by the time the light is received, this component of its velocity has actually moved the body into the position P' where it is seen. Since P' is the apparent position of P, the observed position is also identical with the actual geometric position at t — Δί; the component V@ does not change the geometric direction from the Earth. Meanwhile, however, a further geometric displacement has occurred from the other component VP — V@ of the velocity, which displaces the actual geometric position Q at the time t of observation by an amount Aq from P'. Because of this displacement, the geometric position at the time of observation is in

—►

the direction of the resultant of OP' and VP — V@ ; and to the first order in VIc, the planetary aberration Aq may be obtained by taking the components of the velocity of the Earth relative to the planet, — (VP — V@), for the values ofdx/dt, etc., in the formulas (64) and (65) previously used for stellar aberra-tion.

In rectilinear uniform motion, a body which was at the point P(x0, y0, z0) in space at time t — At will have moved to the geometric position

x = x0 + x At, y = y0 + y At, z = z0 + z Δί,

at the time of observation t. The component of its velocity that is equal to K@ moves it to the geometric position

x = x0 + x@ Δί, / = y0 + y@ Δί, z = z0 + z@ Δί;

since this is the same as the apparent position at time t, the planetary aberra-tion to the first order is represented by

x' - x = (x@ - χ) At,

y -y = (y@-y)&t, (81)

To this order of accuracy, the angular displacement kq of the apparent position from the geometric position at the same instant is equivalent to a linear geometric displacement in space, relative to the Earth, of amount Δ^

represented by the left-hand members of (81). The right-hand members represent the motion of the Earth relative to the body during the light time Δ/, i.e., the negative of the geocentric motion of the body in space. The relative velocity V@ — VP is at an angle q with the line of sight; the dis-placement Δ</ is produced by the component of this relative motion perpendi-cular to the line of sight, (K@ — VP) sin q at. At the geocentric distance

PLANETARY ABERRATION 133 p = c Δί, the linear displacement that subtends an angle Δ# at the Earth is p Δ</, and therefore

Λ^v® -^VP ·

Ag = — sin <?,

exactly analogous to the first-order term in the expansion of (61).

In the equatorial system, the rectangular geocentric coordinates are

ξ = p cos ô cos α, η = p cos ô sin a, ζ = p sin <5. (82) In terms of these coordinates, (81) becomes

Differentiating (82) with respect to a and ô gives equations for the com-ponents —Δ£, —Δη, —Δζ of Δ$; solving Δ£ and Δ77 for cos δ Δα and sin ô Δό, and Δζ for cos ô Δ<5, gives for the planetary aberration in right ascen-sion and declination

(a — a) cos 0 = sin a cos a — , P P

Δ£ Δι? Δζ ό' — ό = cos a sin ό h sin a sin δ cos ô — ,

(83)

Since

^ = + 1 ? , . . . (S4>

p c at

the equations (83) are the same as the first-order terms of Eqs. (64) and (65) with the relative velocities of the Earth —d£\dt, —αη/ώ, —dljdt in place of the total velocities.

Rigorously, the light time Δί in the aberration formulas is the interval of time required for the light to traverse the distance from the point in space where the body is located at time t — Δ/ to the position of the observer at time t\ but this distance cannot be determined until Δί is known, and in practice the light time may ordinarily be calculated from the geocentric distance of the body at either t or t — Δί without appreciable error. Successive approximations may be used if greater precision is required.

For the purpose of practical calculation, the. planetary aberration may be expressed directly in terms of the relative motion to which it is entirely due.

The geocentric velocity of the body perpendicular to the line of sight is -(V@ -VP) sin q = R^d-^

where dqjdt is the consequent angular velocity of the apparent motion on the celestial sphere; the left-hand member is — c Aq, where Aq is the displacement of the apparent position P' from the geometric position P at the same instant, in the direction of +(K@ — VP). The amount of the displacement Αφ in any apparent coordinate φ on the celestial sphere is the corresponding component of Aq and is in the direction opposite to dop\dt\ therefore

c dt dt

from which either <p(P') or φ(Ρ) may be found from the other, since dy\dt in the right-hand member may be evaluated by numerical differentiation of a geometric ephemeris computed from gravitational theory.

The geometric position φ(Ρ) at the time of observation is obtained by adding to the observed position the amount of apparent motion during the light time. Conversely, the apparent position at any time t may be obtained by subtracting from the geometric position at time / the motion during the light time, or, equivalently, interpolating in a geometric ephemeris from / to t — At, in accordance with the principle that the observed position at time / is the same as the geometric position relative to the Earth was at time t — At.

Likewise, in accordance with this principle, instead of correcting an observed position to obtain the geometric position at the same instant, the time of

observation may be antedated to t — At, to obtain the geometric position at this previous instant.

Accordingly, in the practical calculation of geocentric ephemerides of apparent planetary positions, the method that has usually been followed to obtain the ephemeris position for a tabular time t is to add to the geometric position at time t in each coordinate φ as computed from gravitational tables the correction

498⁸.38Û , . . . . Λ ,

- x (motion in w in 2 days) 60 x 60 x 24 x 2

= —0.0028841/9 x (motion in 9? in 2 days),

which is the negative of the geocentric motion in this coordinate during the light time; 498⁸.38 is the adopted value of the time required for light to travel unit distance, corresponding to a constant of aberration 20".47.

PLANETARY ABERRATION 135 The two-day motion is obtained by subtracting the geometric coordinate one day before t from the value one day later than /.

The formulas (67) and (83) for aberration are of the same form as the formulas for annual parallax, with the distance of the body replaced by the velocity of light, and the distance between the two points of observation replaced in stellar aberration by the negative of the velocity of the observer, in planetary aberration by the negative of the relative velocity. Planetary aberration is therefore sometimes called the parallax of light.

App.pos.

time t True pos. \

t-Ah \

\ V 0(xt,yt,zt) " ~ \ ^

/Orbit of the earth

FIG. 25. Planetary aberration in curvilinear motion. The apparent position is in the direction ET, where P is the geometric position of the object at time t — Δ/, and E' is the position which the Earth would have occupied at / — Δί if it had been moving during Δ/ with the same velocity as it instantaneously has at time t.

The preceding methods may be appreciably in error when the light time is large, since they depend upon the hypothesis that the Earth and the planet both are in uniform rectilinear motion. Actually, the geometric position of

In document CHAPTER 6 Aberration (Pldal 26-46)