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Note on the influence of dark matter on the motion of the solar system
Physics I.¢tters A 184 (1993) 41-44 North-Holland
PHYSICS LETTERS A
Note on the influence of dark matter on the motion of the solar system Sergei Klioner 1 National Astronomical Observatory, Mitaka, Tokyo, Japan and Michael Soffel UniversiRit Tiibingen, Tfibingen, Germany Received 21 July 1993; revised manuscript received 1 November 1993; accepted for publication 4 November 1993 Communicated by J.P. Vigier
The influence of dark matter on the motion of the solar system is reinvestigated. The influence is proved to be several orders of magnitude smaller than effects that are detectable at the present time.
In a recent Letter Braginsloy et al. [ 1 ] have studied the influence of dark matter on the motion of the planets and satellites in the solar system. They came to the conclusion that such an influence might be detectable, e.g., in the motion of jupiter, in the foreseeable future. However, their approach is not strictly justified (e.g., they did not account for the correction to r -3 in their equation (5)). This motivated us to look again into this problem from a more rigorous point of view. We reinvestigated their problem for the motion of the solar system by considering the Keplerian two-body problem (sun and a planet) with two different perturbing potentials, V1 -----½/~1x2--~- ½/~ir282,
(1)
V2 ----½?/2(3x2- r 2) = ½?12r2(3s 2 - 1 ) ,
(2)
where r=a(1 - e 2 ) / [ 1 + e c o s ( f ) ], s=cos(Q) c o s ( t o + f ) - s i n ( ~ ) s i n ( t o + f ) cos(i),
(3)
and a, ~2, to, i being the usual Keplerian elements of the planetary orbit a n d f i s the true anomaly. VI corresponds to the form of acceleration assumed by Braginsky et al. [ 1 ] (the x-axis pointing in the direction of the dark matter potential gradient) if
01 =4~Gpd= ~ GMd(a) ,
(4)
where #d is the density of the dark matter and M a ( a ) = ~rcaapdis essentially the mass of the dark matter contained in a volume that is determined by the semimajor axis of the planetary orbit. V2 corresponds to the first order term of the tidal influence of a third perturbing body situated on the x-axis. Assuming the cloud of clark On leave from: Institute of Applied Astronomy, Saint Petersburg, Russian Federation. Elsevier Science Publishers B.V.
41
Volume 184, number 1
PHYSICS LETTERS A
27 December 1993
matter to be approximately spherical with respect to the galactic center located at a distance a' from the sun we take r/2 -- G M da( , ~a ' ) = ~ n G p d ~ r/l
(5)
The result r/l ~ r/2 justifies the use of I"1 for order of magnitude estimates. To derive secular perturbations of first order we average VI,2 over one complete revolution of the planet, 2~x Vi = ~
Vi dM
(6)
,
o
17I = ½r/la2(½0t- l e 2 f l ) ,
(7)
¢2 = - ½r/2a2( 1 + ~e 2 - ~ a + ~e2#),
(8)
where tx=cos2(t2) + c o s 2 ( i ) sin2(I2), fl= ~ cos(i) sin(2to) sin (2~2) + cos2(t2) [ 1 - 5 cos 2 (09) ] + e o s 2 ( i ) sin2(t2) [ 1 - 5 sin2(to) ] .
(9)
From ( 7 ) - ( 9 ) the desired first-order perturbations are directly obtained from the Lagrange planetary equations (see, e.g., ref. [ 2 ] ). In the following we present first order secular variations of the elements per revolution: (Aa)rev=0,
(10)
(Ae)r~v - 5nrh ex/1 2n 2 - e 2 {cos(i) sin(212) cos(2o9) + [cos2 (t2) - c o s 2 ( i ) sinE(O) ] sin(2tO)},
( 11 )
xrh (Ai)r~ = 2n2 lx/q-~_e2 sin(i) (sin(212)-e2{sin(212) [ 1 - 5 cos2(oo)] + 5 cos(i) sin(2og) sin2(£2)}),
(12)
~rh (A/2)~v = - 2n2 lx/~-~_e2 (2 cos(i) sinE(Q) --e2{~ sin(2og) sin(2t2) + 2 cos(i) sin2(I2) [ 1 -- 5 sin2(og) ]})
(13)
(the estimates in ( 1 0 ) - ( 13 ) are for lit. To obtain the values for V2 these estimates should be multiplied with a factor 3, and r/1 should be replaced by r/E), (Aog)re~ +Cos(i) (AI2)r,v =
-(AMo)rev = -
=-
for/Ix/1 n2 -- e2 fl, nr/2 N/1
- e2
n2
2xr/----L [t~-½(l+e2)fl] n 2
for Vi ,
(3+3fl),
for I/'2, for I/1 ,
,
2Xr/e { 3 [ o t _ ½ ( l + e 2 ) f l ] _ ½ ( 7 + 3 e e ) n----T-
(14)
}
for V2.
(15)
The secular terms in co, t2 and Mo give a constant change in the mean motion and, therefore, linear drift in the mean longitude of the planet, (A2)r~v = (AMo)r¢~ + (Aco)~v + (Afd)~v 42
(16)
Volume 184, number 1
PHYSICS LETI'ERS A
27 December 1993
per unperturbed period. The third-body perturbations coincide with ref. [3 ] except for (AMo)r~v which has not been considered there. Note that the order of magnitude for a change of some orbital elements e per revolution is just given by (A~)~v~th,2n-2,.,Md(a)/Mo. Considering that 0
(17)
1 P r/ arcsec <8"2X 10-9 x/l'e2-- 1 yr I0-3° s-2 century '
( 18 )
I (A.Q)~v [ < 1 × 1 0 -a ~ ~P r/ arcsec x/1 _e2 l yr 10 -3° s -2 century'
(19)
P !/ arcsec I (AtO)r,~ + c o s ( i ) (Al2)~v I < 6.4 × 10 -9 1 yr I0 -3° s -2 century'
(20)
P r/ arcsec I ( A 2 ) ~ I < [ 2 8 + 2 5 ( 1 - e 2) -1/2] × 8.2 × 10 -1° 1 yr 10 -30 s -2 century'
for 1/1,
P r/ arcsec < [98+75(1 - e 2) -1/2] × 8 . 2 × 10 -1° 1 yr 10 -30 s -2 century'
for V2.
(21 )
where P is the orbital period of a planet. The estimates in (17 ) - ( 2 0 ) are for VI. To obtain the values for V2 the estimates in ( 17 ) - (19) should be multiplied with a factor 3, and that in (20) with a factor 9. We see that the effect is proportional to the body's period. Even for pluto ( P ~ 2 4 8 yr, e=0.25) one gets arcsec I (A2)r~ [ < 3 . 6 × 10 -5 i0_3~ S--2 century"
(22)
The maximal possible value of t/is 10-29 s-2. This results from the maximal possible value of the mass of the halo being 10~2Mo [ 4 ] and an assumption that all the mass is concentrated within a sphere of radius of 8 kpc). Braginsky et al. [1] used 17=4.5X10-als -2 (pd=O.3GeV/cm3=5.3×lO-25g/cm3). The estimate pa=O.OlOMs~/pc3 taken from ref. [4] gives p d = 6 . 8 × 10-2Sg/cm 3, and r/= 1.9× 10 -31 s -2. The obtained numerical estimates should be compared with the modern accuracy of the planetary ephemeris. According to refs. [ 5,6 ] the errors in mean motions of the major planets are larger than 0.01 arcsec per century (2 aresec per century for pluto). Accurate determination of the inertial mean motion of planets is complicated by the uncertainties of the masses of the planets and asteroids. Although post-fit residuals of ranging observations of the Viking lander are 7-12 m, that is (3-6) × 10-6 arcsec, even at the middle of the data span, the uncertainty of the inertial mean motion of mars is 0.003 arcsec per century [ 6 ]. Therefore, the possible influence of the dark matter is 2-3 orders of magnitude smaller than the effects detectable now. Let us note that the problem of the influence of outer matter (nearest stars and the galaxy as a whole) has been considered in classical celestial mechanics many times (see refs. [ 7,8 ] for reviews). The answer was that the influence can be neglected for the motion of the planets, but might be significant for long periodic comets whose aphelia are comparable with the distance to the nearest stars. We also should mention that for the galaxy as a whole considered to be spherically symmetric 17~1.8×10-3°s -2 and for the nearest star otCen t/~, 3.8 × 10-3°s-2 (the latter estimate is derived from the values of the mass of M,~c,~ = 2Mo and distance to the sun R~c,~ = 1.33 pc taken from ref. [ 9 ] ). Therefore, the tidal influence of the galaxy and nearest stars is anyway several times larger than the possible influence of the dark matter. Besides that, the perturbations due to the dark matter do not contain in the mean longitude terms quadratic with respect to time (secular acceleration) and, therefore, cannot be interpreted as a time variation of the gravity constant G. 43
Volume 184, number I
PHYSICS LETTERS A
27 December 1993
S.K. w o u l d like to t h a n k t h e J a p a n S o c i e t y o f t h e P r o m o t i o n o f S c i e n c e for f i n a n c i a l s u p p o r t .
References [ 1 ] V.A. Braginsky, A.V. Gurevich and K.P. Zybin, Phys. Lett. A 171 (1992) 275. [2] A.E. Roy, Orbital motion (Hilger, Bristol, 1982). [3] G.E. Cook, Cmophys. J. RAS 6 (1962) 271. [ 4] K.C. Freeman, in: Dark matter in the universe, eds. J. Kormendy and G.R. Knapp (Reidel, Dordrecht, 1987) p. 119. [ 5 ] E.M. Standish and J.G. Williams, in: Inertial coordinate system on the sky, eds. J.H. Lieske and V.K. Abalaldn (Kluwer, Dordrecht, 1990) p. 173. [6] J.G. Williams and E.M. Standish, in: Reference frames, eds. J. Kovalevky, LI. Mueller and B. Kolaczek (Kluwer, Dordrecht, 1989) p. 67. [ 7 ] G.A. Chebotarev, in: The motion, evolution of orbits, and origin of comets, eds. G.A. Chebotarev et al. (Reidel, Dordrecht, 1972 ) p. 1. [8] M.V. Torbett, in: The galaxy and the solar system, eds. R. Smoluchowski, J.H. Bahcall and M.S. Mattbews (University of Arizona Press, Tucson, 1986) p. 147. [9] P. Demarque, D.B. Guenter and W.F. van Altena, Astrophys. J. 300 (1986) 773.
44
PHYSICS LETTERS A
Note on the influence of dark matter on the motion of the solar system Sergei Klioner 1 National Astronomical Observatory, Mitaka, Tokyo, Japan and Michael Soffel UniversiRit Tiibingen, Tfibingen, Germany Received 21 July 1993; revised manuscript received 1 November 1993; accepted for publication 4 November 1993 Communicated by J.P. Vigier
The influence of dark matter on the motion of the solar system is reinvestigated. The influence is proved to be several orders of magnitude smaller than effects that are detectable at the present time.
In a recent Letter Braginsloy et al. [ 1 ] have studied the influence of dark matter on the motion of the planets and satellites in the solar system. They came to the conclusion that such an influence might be detectable, e.g., in the motion of jupiter, in the foreseeable future. However, their approach is not strictly justified (e.g., they did not account for the correction to r -3 in their equation (5)). This motivated us to look again into this problem from a more rigorous point of view. We reinvestigated their problem for the motion of the solar system by considering the Keplerian two-body problem (sun and a planet) with two different perturbing potentials, V1 -----½/~1x2--~- ½/~ir282,
(1)
V2 ----½?/2(3x2- r 2) = ½?12r2(3s 2 - 1 ) ,
(2)
where r=a(1 - e 2 ) / [ 1 + e c o s ( f ) ], s=cos(Q) c o s ( t o + f ) - s i n ( ~ ) s i n ( t o + f ) cos(i),
(3)
and a, ~2, to, i being the usual Keplerian elements of the planetary orbit a n d f i s the true anomaly. VI corresponds to the form of acceleration assumed by Braginsky et al. [ 1 ] (the x-axis pointing in the direction of the dark matter potential gradient) if
01 =4~Gpd= ~ GMd(a) ,
(4)
where #d is the density of the dark matter and M a ( a ) = ~rcaapdis essentially the mass of the dark matter contained in a volume that is determined by the semimajor axis of the planetary orbit. V2 corresponds to the first order term of the tidal influence of a third perturbing body situated on the x-axis. Assuming the cloud of clark On leave from: Institute of Applied Astronomy, Saint Petersburg, Russian Federation. Elsevier Science Publishers B.V.
41
Volume 184, number 1
PHYSICS LETTERS A
27 December 1993
matter to be approximately spherical with respect to the galactic center located at a distance a' from the sun we take r/2 -- G M da( , ~a ' ) = ~ n G p d ~ r/l
(5)
The result r/l ~ r/2 justifies the use of I"1 for order of magnitude estimates. To derive secular perturbations of first order we average VI,2 over one complete revolution of the planet, 2~x Vi = ~
Vi dM
(6)
,
o
17I = ½r/la2(½0t- l e 2 f l ) ,
(7)
¢2 = - ½r/2a2( 1 + ~e 2 - ~ a + ~e2#),
(8)
where tx=cos2(t2) + c o s 2 ( i ) sin2(I2), fl= ~ cos(i) sin(2to) sin (2~2) + cos2(t2) [ 1 - 5 cos 2 (09) ] + e o s 2 ( i ) sin2(t2) [ 1 - 5 sin2(to) ] .
(9)
From ( 7 ) - ( 9 ) the desired first-order perturbations are directly obtained from the Lagrange planetary equations (see, e.g., ref. [ 2 ] ). In the following we present first order secular variations of the elements per revolution: (Aa)rev=0,
(10)
(Ae)r~v - 5nrh ex/1 2n 2 - e 2 {cos(i) sin(212) cos(2o9) + [cos2 (t2) - c o s 2 ( i ) sinE(O) ] sin(2tO)},
( 11 )
xrh (Ai)r~ = 2n2 lx/q-~_e2 sin(i) (sin(212)-e2{sin(212) [ 1 - 5 cos2(oo)] + 5 cos(i) sin(2og) sin2(£2)}),
(12)
~rh (A/2)~v = - 2n2 lx/~-~_e2 (2 cos(i) sinE(Q) --e2{~ sin(2og) sin(2t2) + 2 cos(i) sin2(I2) [ 1 -- 5 sin2(og) ]})
(13)
(the estimates in ( 1 0 ) - ( 13 ) are for lit. To obtain the values for V2 these estimates should be multiplied with a factor 3, and r/1 should be replaced by r/E), (Aog)re~ +Cos(i) (AI2)r,v =
-(AMo)rev = -
=-
for/Ix/1 n2 -- e2 fl, nr/2 N/1
- e2
n2
2xr/----L [t~-½(l+e2)fl] n 2
for Vi ,
(3+3fl),
for I/'2, for I/1 ,
,
2Xr/e { 3 [ o t _ ½ ( l + e 2 ) f l ] _ ½ ( 7 + 3 e e ) n----T-
(14)
}
for V2.
(15)
The secular terms in co, t2 and Mo give a constant change in the mean motion and, therefore, linear drift in the mean longitude of the planet, (A2)r~v = (AMo)r¢~ + (Aco)~v + (Afd)~v 42
(16)
Volume 184, number 1
PHYSICS LETI'ERS A
27 December 1993
per unperturbed period. The third-body perturbations coincide with ref. [3 ] except for (AMo)r~v which has not been considered there. Note that the order of magnitude for a change of some orbital elements e per revolution is just given by (A~)~v~th,2n-2,.,Md(a)/Mo. Considering that 0
(17)
1 P r/ arcsec <8"2X 10-9 x/l'e2-- 1 yr I0-3° s-2 century '
( 18 )
I (A.Q)~v [ < 1 × 1 0 -a ~ ~P r/ arcsec x/1 _e2 l yr 10 -3° s -2 century'
(19)
P !/ arcsec I (AtO)r,~ + c o s ( i ) (Al2)~v I < 6.4 × 10 -9 1 yr I0 -3° s -2 century'
(20)
P r/ arcsec I ( A 2 ) ~ I < [ 2 8 + 2 5 ( 1 - e 2) -1/2] × 8.2 × 10 -1° 1 yr 10 -30 s -2 century'
for 1/1,
P r/ arcsec < [98+75(1 - e 2) -1/2] × 8 . 2 × 10 -1° 1 yr 10 -30 s -2 century'
for V2.
(21 )
where P is the orbital period of a planet. The estimates in (17 ) - ( 2 0 ) are for VI. To obtain the values for V2 the estimates in ( 17 ) - (19) should be multiplied with a factor 3, and that in (20) with a factor 9. We see that the effect is proportional to the body's period. Even for pluto ( P ~ 2 4 8 yr, e=0.25) one gets arcsec I (A2)r~ [ < 3 . 6 × 10 -5 i0_3~ S--2 century"
(22)
The maximal possible value of t/is 10-29 s-2. This results from the maximal possible value of the mass of the halo being 10~2Mo [ 4 ] and an assumption that all the mass is concentrated within a sphere of radius of 8 kpc). Braginsky et al. [1] used 17=4.5X10-als -2 (pd=O.3GeV/cm3=5.3×lO-25g/cm3). The estimate pa=O.OlOMs~/pc3 taken from ref. [4] gives p d = 6 . 8 × 10-2Sg/cm 3, and r/= 1.9× 10 -31 s -2. The obtained numerical estimates should be compared with the modern accuracy of the planetary ephemeris. According to refs. [ 5,6 ] the errors in mean motions of the major planets are larger than 0.01 arcsec per century (2 aresec per century for pluto). Accurate determination of the inertial mean motion of planets is complicated by the uncertainties of the masses of the planets and asteroids. Although post-fit residuals of ranging observations of the Viking lander are 7-12 m, that is (3-6) × 10-6 arcsec, even at the middle of the data span, the uncertainty of the inertial mean motion of mars is 0.003 arcsec per century [ 6 ]. Therefore, the possible influence of the dark matter is 2-3 orders of magnitude smaller than the effects detectable now. Let us note that the problem of the influence of outer matter (nearest stars and the galaxy as a whole) has been considered in classical celestial mechanics many times (see refs. [ 7,8 ] for reviews). The answer was that the influence can be neglected for the motion of the planets, but might be significant for long periodic comets whose aphelia are comparable with the distance to the nearest stars. We also should mention that for the galaxy as a whole considered to be spherically symmetric 17~1.8×10-3°s -2 and for the nearest star otCen t/~, 3.8 × 10-3°s-2 (the latter estimate is derived from the values of the mass of M,~c,~ = 2Mo and distance to the sun R~c,~ = 1.33 pc taken from ref. [ 9 ] ). Therefore, the tidal influence of the galaxy and nearest stars is anyway several times larger than the possible influence of the dark matter. Besides that, the perturbations due to the dark matter do not contain in the mean longitude terms quadratic with respect to time (secular acceleration) and, therefore, cannot be interpreted as a time variation of the gravity constant G. 43
Volume 184, number I
PHYSICS LETTERS A
27 December 1993
S.K. w o u l d like to t h a n k t h e J a p a n S o c i e t y o f t h e P r o m o t i o n o f S c i e n c e for f i n a n c i a l s u p p o r t .
References [ 1 ] V.A. Braginsky, A.V. Gurevich and K.P. Zybin, Phys. Lett. A 171 (1992) 275. [2] A.E. Roy, Orbital motion (Hilger, Bristol, 1982). [3] G.E. Cook, Cmophys. J. RAS 6 (1962) 271. [ 4] K.C. Freeman, in: Dark matter in the universe, eds. J. Kormendy and G.R. Knapp (Reidel, Dordrecht, 1987) p. 119. [ 5 ] E.M. Standish and J.G. Williams, in: Inertial coordinate system on the sky, eds. J.H. Lieske and V.K. Abalaldn (Kluwer, Dordrecht, 1990) p. 173. [6] J.G. Williams and E.M. Standish, in: Reference frames, eds. J. Kovalevky, LI. Mueller and B. Kolaczek (Kluwer, Dordrecht, 1989) p. 67. [ 7 ] G.A. Chebotarev, in: The motion, evolution of orbits, and origin of comets, eds. G.A. Chebotarev et al. (Reidel, Dordrecht, 1972 ) p. 1. [8] M.V. Torbett, in: The galaxy and the solar system, eds. R. Smoluchowski, J.H. Bahcall and M.S. Mattbews (University of Arizona Press, Tucson, 1986) p. 147. [9] P. Demarque, D.B. Guenter and W.F. van Altena, Astrophys. J. 300 (1986) 773.
44