Smithsonian Contributions to Astrophysics VOLUME 5, NUMBER 9 ROTATION OF AN EARTH SATELLITE IN FLIGHT ALONG ITS ORBIT By YUSUKE HAGIHARA SMITHSONIAN INSTITUTION Washington, D.C. 1961 Publications of the Astro physical Observatory This series, Smithsonian Contributions to Astrophysics, was inaugurated in 1956 to provide a proper communication for the results of research con- ducted at the Astrophysical Observatory of the Smithsonian Institution. Its purpose is the "increase and diffusion of knowledge" in the field of astro- physics, with particular emphasis on problems of the sun, the earth, and the solar system. Its pages are open to a limited number of papers by other investigators with whom we have common interests. Another series is Annals of the Astrophysical Observatory. It was started in 1900 by the Observatory's first director, Samuel P. Langley, and has been published about every 10 years since that date. These quarto volumes, some of which are still available, record the history of the Observatory's researches and activities. Many technical papers and volumes emanating from the Astrophysical Observatory have appeared in the Smithsonian Miscellaneous Collections. Among these are Smithsonian Physical Tables, Smithsonian Meteorological Tables, and World Weather Records. Additional information concerning these publications may be secured from the Editorial and Publications Division, Smithsonian Institution, Washington, D.C. FRED L. WHIPPLE, Director, Astrophysical Observatory, Smithsonian Institution. Cambridge, Mass. For sale by the Superintendent of Documents, U.S. Government Printing Office Washington 25, D.C. - Price 30 cents Contents Pace Introduction 113 Equations of motion 113 Equatorial circular orbits 119 Inclined circular orbits 124 Eccentric orbits 134 Summary 139 Keferences 142 Abstract 143 in Rotation of an Earth Satellite in Flight along its Orbit By YUSUKE HAGIHARA 1 Introduction The translational motion of an earth satellite as affected by radiation pressure (Musen, 1960), the electrostatic charge induced in the satellite body by the charge distribution in space, the electric current induced by the satellite's motion through the geomagnetic field (Jastrow and Pearse, 1957; Wyatt, 1960; Beard and Johnson, 1960), and the induced magnetization caused by the geomagnetic field (La Paz, 1960) have been discussed recently. The rotational motion of an earth satellite, however, has not yet been worked out in detail, except for the work of Roberson (1958) and others on the gravitational torque. In the present paper I propose to discuss the rotational motion of an earth satellite during the period of its flight along its orbit around the earth's center, and shall base the discussion on Euler's equations of motion for the rotation of a rigid body around its center of mass. The body of the satellite is supposed to be symmetrical, not only in its external shape but also in its dynamical properties about its axis of symmetry; that is, if we denote the principal moments of inertia by A, B, C, then we have A?B. Although it is assumed that C2, and ?3. Then Euler's equations of motion for the rotation of the satellite around its center of mass (see, for example, Klein and Sommerfeld, 1923; Whittaker, 1927; Charbonnier, 1927; Deimel, 1950) are: (1) where Kia1} K2O2, K3O}3 represent the rate of retardation of the angular velocities around the three principal axes of inertia due to any cause. Take coordinate axes of fixed direction in space with the center of mass of the satellite as the origin; for example, the equatorial axes for which the Z-axis is directed parallel to the earth's north FIGUU 1.?Relation between the coordinate system (XYZ) fixed in space and the routing coordinate system (xyx) attached to the satellite, with the Eulerian angle NO. ? ROTATION OF AN EABTH SATELLITE 1 1 5 pole, the JY-axis in the plane of the earth's equator and directed to the vernal equinox, and the Y- axis in the plane of the earth's equator and perpendicular to the JST-axis. Denote the Eulerian angles of the principal axes of inertia of the satellite referred to these (-XTZ)-axes by 0, i=-j-sin^?-j*rsin0cos^, a>2=-^cos^+-^-sin0sin^, ?*==?t^~dtC0S ' ^ or wi sin ip-\- w2 cos \p =-n> ? ?i cos ^+a>2 sin $=jf: sin 0. (2a) ^ The force function for the tide-generating action on the satellite is expressed (Plummer, 1918) by the equation I^^'-]?' (3) where mr denotes the mass of the earth, G the constant of universal gravitation, a the semimajor axis of the satellite's orbit, r the radius vector of the satellite, both counted from the earth's center, and z is the rectangular coordinate in the direction of the principal axis C of the satellite. Write C-A 3m'GA E A ?? 2a3 ? e? then we have, Here the (xyz) -axes are the moving coordinates attached to and carried by the satellite, and their directions coincide with its principal axes of inertia. By substituting equation (2) in equation (1) and noting that . . bU , . W W . ZU . , W W VU W s i n v ^ ? r c o s ^-^?=^?> c o s ^ ^ sin ^-5?=?5?1 3r~:saE3rr?Y dx dy d6 dx dy &p dz d^ we get the equations of motion (1) in the form: A ^ A (W - ? ?? " + c l - > (?M ???) =?KI sin ^ (-r: sin \p? ~? sin 0 cos ^ j?JC2 COS ^(JI cos ^+^7 sin 0 sin ^ )+-^? . ,fdd ,.d

K3. From equations (1) we have, by neglecting the force function U, From the third equation of (5) we get io3=a exp*j ? -gt >, (6) where a is an integration constant. Inserting this expression in (5) we have From these we obtain the equations: Integrating these, we have " | ^ ? y ^ t [ ^ exp | - with two integration constants 5 and i?. From these equations we can see that, if initially a>i=a>2 = 0 , then ajj and o>2 are always zero, and that o>3 varies more rapidly than o>i and OJ2, as Cs sin 8S s m >^s cos 03 By substituting equations (6) and (7) in these ratios we get ? , (8) tan 0,=^ exp / ( _ | + ? ) ^ . (9) We can see from these that the polar angle 6, of the instantaneous axis tends to TT/2 for f->?? if (K,/.A)< (K3/<7) ; and tends to 0 for t-^co if (KI/A)XK3/C). The azimuth angle a immersed in an incompressible in viscid fluid of density p. In a two-dimensional irrotational motion of the fluid across the ellipse with ROTATION OF AN EARTH SATELLITE 117 FIGURE 2.?Two-dimensional stream lines around an ellipse immersed in an incompressible fluid. speed V at infinity, while making an angle 5 with the major axis of the ellipse, the torque M exerted to the ellipse is, according to Cisotti (Lamb, 1932; Milne-Thomson, 1955), in the sense of increasing 8, or \M\=irP(<*?a2) Vs sin 8 cos 8, \M\=M sin 5 cos 8, M=TTP{&?a2) V2. (10) (11) We extend this result to a three-dimensional motion for an ellipsoid of revolution c > a = b . Then the same formula (11) can be expected to represent the torque exerted on the ellipsoid moving with speed V relative to the fluid by making an angle 8 with the major axis c of the ellipsoid. In the high atmosphere the air density is so low that this hydrodynamic approximation is certainly not valid. Until a rigorous aerodynamical treatment is available, based on a consideration of the detailed mechanisms involved when particles of the atmosphere impinge on the surface of the satellite, we shall tentatively assume the formula (11), with a correcting numerical factor (Rand Symposium, 1959) which I shall omit.2 There may be a torque due to radiation pressure. If the resultant R of the solar radiation pressure reacts at a point at a distance D from the center of mass of the satellite in the direction of its longest axis, which makes an angle +2 sin (v+w) cos (?+?) sin1 d sin

and # i > 0 for CA. The solution of (22) is 0=0O= constant, p?) + l], (24) where v=ix (t?tx), if we denote the mean motion by n and the epoch of the pericenter passage by t\. The 2-axis of the satellite makes a nutation with the period of the orbital revolution of the satellite superimposed on this precessional motion. Hydrodynamic torque.?According to equation (11) the hydrodynamic torque M is represented by | M | = M sin 8 cos 5 in the sense of increasing 5. We first compute the angle 8. For a circular motion of the satellite we have, from equation (20), =an sin (w-K>), -^ -=?a^ cos By referring to table 1, we find the angle 5 between the 2-axis of the satellite and the direction of the relative velocity V of the atmospheric gas and the satellite, with V=|V|: cos as 5= ?77^ sin 0 cos )cos0, m?? sin (o+w) cos0, n=sin (t?+oj)sin 0sin^+cos (?+w)sin0 cos^>. (28) NO. ? ROTATION OF AN EARTH SATELLITE 121 The directions of the vector M referred to the (xyz)-&xes attached to the body of the satellite are, from table 1 (p. 142), \ = l(x)X)-\-m(x,Y)-\-n(x,Z) = ?cos (y+o>)(cos tp cos ^?cos 0 sin

) (sin

p), My/\M\=l(y,X)-{-m(y,Y)+n(y,Z)=cos (y+a>) (cos

) sin >^, Thus we have the part due to the hydrodynamic torque in the equations of motion by substituting these expressions together with (11) and (26) in equation (17): (-77 ) =?M sin 0 cos 9 sin2 (y+w??)+l]+M sin (y+co?) Vl?sin1 0 sin2 (y+co?p). (30) at Take the mean values of the right-hand members with regard to t, that is, with regard to v from 0 to 2TT, since we are dealing with the case of circular motion; then (sin 2(y+co?) Vl?sin2 0 sin2 (y+w?^))=0, where F and E, respectively, denote the complete elliptic integrals of the first and the second kinds (de Haan, 1858): /?r/2 fr/2 F (sin 0) = Vl-sin2 0 sin2 x ? dx, E (sin 0)= , Jo Jo VI?a 2 0 sin2 z 1 2 2 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS Thus we get the equations of motion: do ,., ? n n f2sin20? MsinecoBd\ (8in w * / ^ (31) In order to solve these equations we expand the elliptic integrals in powers of sin2 0, by assuming that sin 6 < 1: E o*?-i The first equation of (31) takes the form dd sin 5 cos 5 ri i . , . , \\2 4 ? /Mdt After integration for a small sin2 0 we get approximately, tan0=tan0 o -exp/ ?^-(t-t0) \ , (32) where we suppose that 0=0O for t=to. By substituting (32) in the second equation of (31) and integrating it we get, approximately, ^ ( - . . .) (t-Q, (33) where we take ^.(It Clt In order to solve these equations we suppose that sin 0^0, cos ^ dt or, sin 0 ? cos ) sin ft+sin (#+?) cos ft cos / } , ZB = ?a sin (v-\-w) sin / . (38) Thus we have as before, z=XE sin 0 cos i?3* & c o s 2 J+sin2 ft) sin20 sin2

by substituting the values of d0/dt and d

, ZB=?a sin (v-\-ui), 2 \ 2 \ 1 1 1 - I ) = - cos2 Q> s in 2 0 cos2 s in 2 8 s in2 ) cos ft cos / , na at 1 d ZB / 1 \ ? r 5? =cos (v+w) sin / , p.a at We get from equation (27), page 120, the equations: Z={sin (v-\-w) sin ft?cos (#+?) cos ft cos /}cos 0+cos (w+w) sin / s in 6 sin ^ ?{sin fy+co) sin ft?cos (tf+co) cos ft cos /} sin 6 cos sin ^+sin

) cos (ft?v?) cos 0?cos (v+a>){sin (ft? )+cos (?+w) cos (ft?v) cos/] ,\dt / 2 sin 0 (-J7 ) = M sin 5 cos 5 ? sin (u+a>) sin (ft ? ), sin P2 sin ZZPz=sin 0 cos (ft ?*?), we see, by referring to figure 3, that cos 8=?sin Pz cos (#+&>+ /ZPz) and that -~=?Ei sin 0i ? sin 2(tf+a>?^)? Msin 0i cos0i sin2 (w+w?^) Vl~sin20! sin2 ^=Ey cos 0i ? [cos 2(fl+u?^>i)+l]+Msin (p-f^?^j) c o s (o+w?i= ^ZPz. Hence the equations can be treated in the same way as in (30), by taking the pole of the orbital plane as the new Z-axis fixed in space for referring the rotational motion of the satellite. If an inclined orbit is given, we obtain the pole of the orbital plane by the equations, XP=sin / s i n ft, YP=? sin 7cos ft, ZP=cos 7, (47) and we study the rotational motion of the satellite by referring to the orbital plane and its pole. Motion of the orbital plane.?We now examine the effect of the motion of the orbital plane on the rotation of the satellite. I t is known that the disturbing action causes the orbital plane of the satellite to rotate around the pole of the equator with a period of a few months (Y. Kozai, per- sonal communication), while the orbital inclination is kept constant. 128 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL. 5 From the spherical triangle PZz shown in figure 3, where we write the angle /.ZPz=> (32a) sin 0 sin (? Q,o? vt)=sin 0iO sin (piO+2?it) ? exp? ? ^ (??10) V- By solving these equations we obtain the angles 0 and r ? r M 1 1 ? tan dp? 0,0? vt)? sin 0iocos70 cos (y>io+?'O + cos0iosin7o-exp-< ? (t? t0) > /[sin0iosin (+3 sin ft cos ft cos X cos 0 jEj [Hi "" Vl+3 sin2 ft " ' L z HXT^?HrXz_3 sin ft cos ft sin 9 sin (X?y). |L| |H| ~~ Vl+3 sin2 ft U 3sinftcosft m ( } c o g o g c Q s ( x _ } s b l-3sin2ft= g m |L| Vl+3 sin2 ft Vl+3 sin2 ft Lj,==3smft_cosft [ g i n ( x _ ^ g m + c o g fl ( x _ ^ c Q s ] + 1-3 sm 2? g m c o g |L| Vl+3 sin2 ft Vl+3 sin2 ft Lz 1?3 sin2 ft . . - ? /v \ 7T1 = -T = s m ? c o s ^ s i n (X?Wi|L| Vl+3 sin2 ft Le 3 sin ft cos ft . ^ N , 1?3 sin2 ft .4 = M cos 0 cos (X?y)+-^ sm 0, Vl+3 i 2 ftIM Vl+3 sin 2 ft L?_3 sin ft cos ft sin (X?). (52) dty3 \aj Because of the orbital motion of the satellite the longitude X and the latitude ft, indicating the position of the satellite, vary according to the relations, sin ft=sin / s in (c-j-w), tan X=cos / t a n (#+?)? (53) 130 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL. B Put . ? _. 3 sin/3 cos/3 . , ? ? . 1?3 sin2/3 v . . . . cos (H, R ) = ? . -=sin Xi, cos (H, Z)= . K =cos Xi. (54) V V l+3s in 2 ^ Vl + 3sin28 Denote the pole of the instantaneous direction of the vector H by H as shown in figure 4. Then ZH?xi, Hz=A, Zz=6. Further denote the angles AZHz and ZHzZ by 0 and X, respectively, the angle ZHZz being 180?? (X? )+sin xi cos 8, sin A sin 6=sin 8 sin (X? 4 sin XX( cos .A cos XH r?-?-) +sin ^1 sin 0 ? -jr- -77?sm Xi sin A cos 9 ? -JT;dp dt dt or ^ = ~ L o ( - Y (1+3 sin2 /3) sin2 ^1 cos yl+cos 9 ? ^ ^+s in 8 sin X ? ? , (57) at \a / dp at dt . , A dQ A ? _ dX\ dp . . . A _ d\ ,__xsin2 A ? -Tr=cos -4 sm 0 ? -J-T- -37?sm 8 sin A cos X ? -r:? (58)dt dp dt dt Now we can divide the satellite's motion into two parts: I. The part of the motion around the direction H of the geomagnetic field at the position (X,/9) of the satellite, when we suppose that H and thus the satellite are instantaneously at rest; that is, when d\/dt=d&/dt=O. II. The part due to the motion of H caused by the translational motion of the satellite in longitude and latitude; that is, when dXfdt^O, NO. 9 ROTATION OP AN EARTH SATELLITE 1 3 1 Accordingly, the equations (57) and (58) split into two parts, dt \dt A+\dt Jn dt KdtA^KdtJn such that 0 =-L0 P?j (1+3 sin2/S) sinM cos A, f w / i =0; (59) ( -y- ) =cos A sin 6 ? -rx -r:?sin 6 sin A cos X ? -y' (60) \dt/n dp at dt At first we integrate the equations (59). The solution, obtained in the same way as in pages 122 to 123, is: 1 _ 1 \ _ M V mA sin AQ/ \aj v / v w*4 ?sin A V + o / 6 = 0 O = constant. (61) Or, if sin A e=9o. (61a) Hence the first part of the motion consists hi an exponential shortening of the arc A, that is, the angular distance of the z-axis of the satellite from the instantaneous direction of the geomagnetic field for the point (X,/3), while the arc A is kept in the same direction fixed in space, as G=0O (see fig. 4). The integration of the second part of the equations is difficult and I do not go into it at present. As the evaluation of the mean values with respect to t of the right-hand members of equation (56) is also difficult, I am now compelled to assume that sin 0 sin A=cos 0 ? 1 + 3 sin I cot 0 ? (cos

^ sin2 v)z-\-- sin2 / cot2 0 sin2 v \> sin ^1 cos A=sin d cos 0 ? 1+3 sin / ? (cot 6? tan 6) (cos

^ sin v cos ?+sin p sin2 v)2+^ sin2 / ? (cot2 0?1) sin2 v \> sin A cos X=sin 6 ? 1+3 sin / cot 0 ? (cos

Yl ^ f ) ~~~^? \ ) cos 0 ? Ksin / cos H~9 sin2 / ? (cot 0?tan 0)Qsin

) sin20sinV+2 sin (?+?) cos (v-\-u) sin20 sin

1 /I1 /I . 1 . . \ sin2 0 ( s m *cos ^+2sin * sm "rw^w ni 2(l-?2)3'2* Hence the equations for the precessional motion are : ( T ) ~0> The solution is easily found: Hence, for an eccentric orbit, the only modification is of the factor 1/(1?e*)3n in the rate of increase of cos (F,V) = ? >Vl 2 2 o o cos (F, ) = , ? e2cos2w V1 ?? cos2 thus we get the values for the direction cosines I, m, n of equation (27): , -Jl ? e2- c o s w ?/=? j L ?== cos 0, ?y/l?e2 cos2 it sin w . cos 0, ?e2 cos2 . . J i ? e2-cosw . . 2 2 sin ^ sin v + , = sin 0 cos +sin w sin v?), cos g= (sin w cos ?p?^l?e2- cos ? sin -(l?e2 cos2 w)-(l?g cosw)~* v, ? /, ?/sin wcos(o? Vl ? e2? cos?sin??V f /sinwcosv>?ijl?e2-coswsiny>V\ Xsuiecosei , -) Vl?e2 cos2 w sin u sin ^ + y i ? e 2 ? cos w cos

) ? -Jl (l-sin? , sin' *)??fc=f ? S (-1)" 2-|=i ? g ? | I. ( *) (sin' ?- F(sin ,)= Jf (1-Sin> ? sin= *)-? 3 being around the z-axis having the moment of inertia C. Let the resistance to the rotation be supposed to be KI?I, K2O>2, K3CO3 around each of the principal axes of inertia. We take reference axes (XYZ) fixed in space and refer the position of the principal axes of inertia of the satellite by use of the Eulerian angles 0, a = b . An irrotational motion of an incompressible inviscid fluid of density p, moving across the spheroid with speed V at infinity, exerts a hydrody- namic torque M making an angle 8 with the major axis c of the spheroid. This torque, according to Cizotti (see Lamb, 1932; Milne-Thomson, 1955), is taken to be: |M|=Msin 8 cos 5, M=7rp(c2-a2)F2, (11) in the sense of increasing 8 (see fig. 2, p. 117). I assume this formula for the moment, as we have no rigorous formula available for our actual case. For an equatorial circular orbit the equations for the precessional motion of the satellite are obtained in equation (31). The hydrodynamical torque affects the precession only in 0, but not in ip. The equations are solved for sin 0-C1 in equations (32) and (33). The angle 0 decreases exponentially and the motion in the angle

^o=?2-^ ? ? ^ ~> 03) in the sense of decreasing y, where i7= |H | denotes the magnetic field at the satellite position and 7 the angle between the geomagnetic field |H| and the z-axis attached to the satellite, of which the latter is supposed to be the direction of the magnetic induction. The earth is assumed to be a uniformly magnetized sphere and the geomagnetic pole is assumed to coincide with the earth's pole. The geomagnetic field is then that of a dipole and is expressed by equation (14) (p. 119). 142 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS For an equatorial circular orbit the magnetic torque affects only the rotational motion in 0 and yff, but not in sin 0 cos

sin 6 sin