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TIDAL VARIATIONS IN THE DYNAMIC PARAMETERS OF PHOBOS M. BURSA Astronomical Institute, Czechoslocak Academy of Sciences, Prague, Czechoslovakia (Received 13 April, 1988) Abstract. Variations in the dynamic parameters of Phobos have been determined after reaching critical value of the semi-major axis rf = 7247 km at which zero-gravity on the surface of Phobos near the equator will take place. The rate of the variations will increase significantly, e.g., in the tidal energy dissipation by one order in magnitude. The total dissipated mechanical energy during the whole tidal history of the system has been estimated as - 5.5 x 10” kg mz s m2, the total decrease in the second zonal Stokes parame- ter of Phobos as -6.6 x IOW’. The tidal friction dynamics generate a large decrease in the semi-major axis a of the orbit of Phobos (Duxbury and Callahan, 1981) da/dt = -2.68 m cy-‘, and, that is why, the decrease in gravity g on the surface of Phobos and in outer space. Let the gravitation of Phobos be modelled by the phobocentric gravitational constant Gm = 6.6 x lo5 m3 sd2, the second zonal Stokes parameter J$O) = -0.0768 and the second sectorial J$‘) = 0.023 1 (BurSa, 1988). Gravity g at any external point P(p, 4. A) then becomes approximately (squares of J$“), Ji2) neglected) g(p, 4, A) = G” P2 i 1 + 3 0 ao 2[J$o)P$o)(sin 4) + P 2 w2p3 + Js2)P’;?)(sin 4) cos 2A] - - - 3 Gm [ 1 - P’,O’(sin 4)] + (1) +5: ; 3 (>[ P’,O)(sin 4) - f PIZ’(sin 4) cos 2;1 11 ; p, 4, A are the phobocentric spherical coordinates (radius-vector, latitude, longitude, respectively), a, = 12 100 m the mean equatorial radius of Phobos; P$O) (sin 4) = (3/2) sin24 - l/2, P’,‘)( sin 4) = 3 cos’ 4, a = 9 378 500 m (Szeto, 1983); o = n = 2.2792 x lop4 rad s-l is the mean motion of Phobos, GA4 = 42 828.44 x lo9 m3 SK* the areocentric gravitational constant; ;1 = 0 defines the plane of the prime meridian of Phobos containing the mass center of Mars (no librations assumed). Because of the synchronous rotation, o2 = GM a -3. Let P be situated at the equator (4 = 0) and prime meridian (A = 0) of Phobos. Then (1) can be specified as ,,,,0,0)=~[1+3~)2(-1J!o’+3~pi)-~(~)*] (2) and the tidal decreasein it as dg(p. O,O> =9GMpda - -. dt a4 dt’ (3) Earth, Moon, and Planets 42 (1988) 227-232. (0 1988 by Kluwer Academic Publishers.

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Page 1: Tidal variations in the dynamic parameters of Phobos

TIDAL VARIATIONS IN THE DYNAMIC PARAMETERS OF PHOBOS

M. BURSA

Astronomical Institute, Czechoslocak Academy of Sciences, Prague, Czechoslovakia

(Received 13 April, 1988)

Abstract. Variations in the dynamic parameters of Phobos have been determined after reaching critical value of the semi-major axis rf = 7247 km at which zero-gravity on the surface of Phobos near the equator will take place. The rate of the variations will increase significantly, e.g., in the tidal energy dissipation by one order in magnitude. The total dissipated mechanical energy during the whole tidal history of the system has been estimated as - 5.5 x 10” kg mz s m2, the total decrease in the second zonal Stokes parame- ter of Phobos as -6.6 x IOW’.

The tidal friction dynamics generate a large decrease in the semi-major axis a of the orbit of Phobos (Duxbury and Callahan, 1981) da/dt = -2.68 m cy-‘, and, that is why, the decrease in gravity g on the surface of Phobos and in outer space. Let the gravitation of Phobos be modelled by the phobocentric gravitational constant Gm = 6.6 x lo5 m3 sd2, the second zonal Stokes parameter J$O) = -0.0768 and the second sectorial J$‘) = 0.023 1 (BurSa, 1988). Gravity g at any external point P(p, 4. A) then becomes approximately (squares of J$“), Ji2) neglected)

g(p, 4, A) = G” P2

i 1 + 3 0 ao 2[J$o)P$o)(sin 4) + P

2 w2p3 + Js2)P’;?)(sin 4) cos 2A] - - - 3 Gm [ 1 - P’,O’(sin 4)] + (1)

+5: ; 3 (>[ P’,O)(sin 4) - f PIZ’(sin 4) cos 2;1 11 ;

p, 4, A are the phobocentric spherical coordinates (radius-vector, latitude, longitude, respectively), a, = 12 100 m the mean equatorial radius of Phobos; P$O) (sin 4) = (3/2) sin2 4 - l/2, P’,‘)( sin 4) = 3 cos’ 4, a = 9 378 500 m (Szeto, 1983); o = n = 2.2792 x lop4 rad s-l is the mean motion of Phobos, GA4 = 42 828.44 x lo9 m3 SK* the areocentric gravitational constant; ;1 = 0 defines the plane of the prime meridian of Phobos containing the mass center of Mars (no librations assumed). Because of the synchronous rotation, o2 = GM a -3.

Let P be situated at the equator (4 = 0) and prime meridian (A = 0) of Phobos. Then (1) can be specified as

,,,,0,0)=~[1+3~)2(-1J!o’+3~pi)-~(~)*] (2)

and the tidal decrease in it as

dg(p. O,O> =9GMpda - -. dt a4 dt’ (3)

Earth, Moon, and Planets 42 (1988) 227-232. (0 1988 by Kluwer Academic Publishers.

Page 2: Tidal variations in the dynamic parameters of Phobos

228 M. BURL%

numerically, putting p = C = 13 500 m (the largest semi-axis of Phobos), the actual value comes out as

dg(& (-40) dt

= - 1.80 x lop9 m ss2 cy-‘.

Because of (4) zero-gravity at P(& 0,O) should occur during the tidal evolution of Phobos, when

3~(~~=1+3~~(-1,,1+35~2))= 1.259,

i.e. at the critical value a = 5 of the semi-major axis of the orbit of Phobos

2 = 7247 km. (6)

However, (4) # constant, because (da/dtl increases, according to the Lagrange plan- etary equation (e.g., Kopal, 1978) as

ah = 3 397 150 m is the mean equatorial radius of Mars, k’ (k) and E’ (E) are the Love numbers and the phase lag angles of Mars (Phobos), respectively (Yoder, 1981; Szeto, 1983). However, because the rotation period of Phobos has been synchronized with its revolution period, only the radial tides take place and E = 0. Then we get from (7) numerically

kfEf = 6.89 x lo-“. (8)

It is the present value; keeping it constant, daldt can be estimated in dependence on a easily. It is represented by curve ( 1) on Figure 1. E.g., for critical value (6) we get dZ/dt = -11.1 mcy-‘.

Value (8) does not respect the fact that the total observed value used in (8) may be affected by the radial tides which do not change the orbital angular momentum

L = $ [G(M + m)a( 1 - e2)] 1:2,

however, they generate the variations in e and a (Goldreich, 1963) conditioned by L = constant:

No data exist for the estimates of the variations in (10) due to the purely radial tides. However, the variations in ( 10) do not contain secular terms (Sidlichovskjr, 1985) and they do not influence the actual secular variation in L practically: namely,

dL L da e de -=-cosi--Lcosi--- dt 2a dt 1 - e2 dt

L sin i di dt’ (11)

Page 3: Tidal variations in the dynamic parameters of Phobos

TIDAL VARIATIONS IN THE DYNAMIC PARAMETERS OF PHOBOS 229

Fig. I. Increase in ( -da/dt) (I) and in (-dH/dt) (2) in dependence on decreasing semi-major axis of Phobos.

With deldt = - 1.02 x lop8 cyy’, di/dt = 1.27 x lop9 cyy’ (Kaula, 1964), i = 1.02’ (inclination), L = 1.982 x 1O26 kg m2 s-- ’ the numerical value valid at present of the total variation in the orbital angular momentum comes out as

dL/dt = -8.98 x lo9 kg m2 s-l. (12)

It is mainly due to da/dt, the contributions by de/dt and dildt are about three orders smaller in magnitude. The corresponding increase in the angular velocity o’ of rotation of Mars comes out as

do’ldt = -dLldtlC’ = 3.23 x 1O-27 rad ss’; (13)

C’ = 2.779 x 1O36 kg m’ is the principal moment of inertia of Mars. Value (12) is about two orders smaller than the decrease (opposite in sign) in o’ due to the solar tides on Mars.

At the critical semi-major axis of the orbit of Phobos (6), the decrease in the angular momentum of Phobos as well as, the increase in the angular velocity of Mars rotation, will be about five times larger than present values (12) and (13).

Page 4: Tidal variations in the dynamic parameters of Phobos

230 M. BURSA

Value ( 12) represents the tidal torque ( --N) exerted on Phobos due to Mars. Opposite in sign ( + N), it represents the tidal torque due to Phobos, exerted on Mars:

(14)

The decrease in the mechanical energy E of the system due to the tidal friction is as

dE -==(a’-n)= -1.41 x 106W; dt (15)

(w’ - n) = - 1.5704 x 10e4 is the difference between the angular velocity of the Mars rotation and the mean motion of Phobos. Variation (15) can be integrated, e.g., during the whole tidal history of the system; it can be rearranged as

(GM) “’ ~-~ 1 a3/2 ’ (16)

L,=L+C’o’=constant=1.970~ 1032kgm2sp’. (17)

The semi-major axis can be expressed as a function of time from (7) (Burns, 1972) as

a(t) = {[a(t = 0)] ‘312 - 39 Gm(GM) -‘~2a~5(k’&‘)m}2/‘3; (18)

t is time counted from the starting epoch t = 0 (4.5 x lo9 y B.P.), a (t = 0) = 18 626 500 m, (k&), is the integral mean value of (k’c’) at the time interval t. Substi- tuting ( 18) into ( 16), the total decrease AE in the mechanical energy during the whole history of the Mars-Phobos system can be calculated from

_ (GM) ‘I2 AE(t) = 3Gm2ah5 $ I, - 3Gm3az 7 Z2 - 3Gm2a~‘(GM)“2Z3, (19)

I, = s

’ (k’c’)[a(t)] -’ dt = (k’e’),,, 0 s

or {[a(t = 0)] 1312 -

- 39Gm(GM)-‘12a~5(k’~‘)mt}-‘2!‘3 dt,

Z2 = I

* (k’E’)(l -e2)‘/2[a(t)]-“~2 & = 0

= (k’.s’)nz,2( 1 - e2)z2 s o* {[a(t = 0)] I312 -

(20)

- 39Gm(GM)-‘~2a~5(k’&‘),t)-“~‘3 dt, (21)

Z, = s

’ (k’&‘)[a(t)] -“j2 dt = (k’c’),, 3 0 s

or {[a(t = 0)] 1312 -

- 39Gm(GM)-‘~2a~5(k’.z’)mt}-‘5~‘3 dt; (22)

Page 5: Tidal variations in the dynamic parameters of Phobos

TIDAL VARIATIONS IN THE DYNAMIC PARAMETERS OF PHOBOS 231

(k’~‘),,~,j = 1, 2, 3 and (1 - e*)z* are the integral mean values; a rough estima- tion gives (k’~‘),,~ = (k’e’), = (S), and (1 - e’):” A 0.980. After integrating and rearranging,

AE(t) = -$ (GM)“2([a(t)]“2 - [a(? = O)]“‘} +

+ 0.490 F [a(t) - a(t = O)] -

- ; GMm{[a(t)] ’ - [u(t = 0)] -‘}. (23)

If we put t = 4.6 x lo9 y, the total tidal decrease in the mechanical energy of the system comes out numerically as

AE = -5.5 x 102’ kg m* s-l. (24)

The tidal decrease in Hamiltonian H = - GMm( 2~) - ’ is mainly responsible for (23). However, dH/dt is not constant, its absolute value increases with decreasing a. Its present value is about -2 x lo6 W, at the critical orbit a = Z (6) it will amount to -20 x lo6 W [curve (2) on Figure l] which should be reached in time interval At ( 18)

Ar=l(GM)‘i2 1 1 39Gmijqk’&‘),., (a 13/2 -ay =4.4x lO’y,

if W-3,. 4 = (k’s’) = (8) = constant. The secular Love number k, of Phobos is close to unity, k, = 1.65, and that is why

it may represent a measure of the body-yield-to-centrifugal deformation in the course of its development during the whole history of its evolution (Munk and MacDonald, 1960). If actually k, = constant during the whole history, it makes possible to express the second zonal Stokes parameter

1 C&7; Jp= --k ~ =

3 ’ Gm (26)

as a function of time: i.e.,

(27)

1 (k’& ‘), 5 GM AJ(P’(Q = --k A 3 s (k,E,)m Gm 4(kdt = o)1’3’2 -

- 39 Gm(GM) -“2u~5(k’~‘),t)-6;13 - [u(t = 0)] -‘}. (28)

Equation (28) represents the decrease in J2 co) during the time interval t = 0 (4.6 x lo9 y B.P.) and t = r. The total decrease during the whole history of the system amounts about to

AJ$” = -6.6 x lo-*. 3 (29)

Page 6: Tidal variations in the dynamic parameters of Phobos

232 M. BURSA

(k’s’),. 5 = (k’ s’), = (8) was adopted for the estimate above. It means, at t = 0 the polar flattening u of the nearly hydrostatic model of Phobos was smaller as at present. Because, for the model in question, however if the synchronous rotation of Phobos,

(30)

the figure parameters at the early history of Phobos are as: c( = 0.02357, J$” = -0.00978. At the critical orbit G (6) their estimates are about E = 0.400, .Tp = -0.166.

Conclusions

(a) After reaching critical semi-major axis (6), the rate of change in the integral of energy of Phobos will amount to about ten times larger value as at present. (b) The total dissipated mechanical energy of the Mars-Phobos system during the whole history of its tidal evolution can be estimated as -5.5 x lo*’ kg m* ss*. (c) The secular Love number of Phobos k, = 1.65 assumed to be constant in the course of its development during the whole history of its evolution makes possible to estimate the long-term decrease in the second zonal Stokes parameter. The total decrease during the whole history of its evolution amounts to -6.6 x lo-*

Acknowledgment

The author wishes to thank Dr. M. Sidlichovsky for valuable comments and correc- tions.

References

Burns, J. A.: 1972, Rec. Geophys. Space Phys. 10, 463. BurSa, M.: 1988, Bull. Astron. Inst. Czechosl. 39 (in print). Duxbury, T. C. and Callahan, J. D.: 1981, Aswon. J. 86, 1722. Goldreich, P.: 1963, Mon. Notices Roy. Astron. Sot. 126, 257. Kaula, W. M.: 1964, Rec. Geophqs. 2, 661. Kopal, Z.: 1978, Dynamics of Close Binary Systems, D. Reidel Publ. Co., Dordrecht, Holland. Munk, W. H. and MacDonald, G. T. F.: 1960, The Rotation of the Earth, Cambridge University Press. Szeto, A. M. K.: 1983, Icarus 55, 133. Sidlichovsk?, M.: 1985, Bull. Astron. Inst. Czechosl. 36, 65. Yoder, Ch. F.: 1982, Icarus 49, 327.