Resonance in the motion of Geocentric Satellite due to Doppler shift of Solar radiation

Resonance in the motion of Geocentric Satellite due to Doppler shift of Solar radiation

Md Sabir Ahamad

Department of Mathematics, SNSRKS College, Saharsa-852202, Bihar, India.

Abstract: – This paper is to discuss the effects of Doppler shift of solar radiation on the resonant motion of geocentric satellite. In presence of Doppler shift of solar radiation resonances 1:1, 1: 2, 1: 3, 2 :1, 2 : 3, 3 :1, 3 : 2, 4 :1, 4 : 3, 5 :1 and 5 : 3 occur. Also discuss the amplitude and time period of the geocentric satellite at all these resonant points.

Key Wards: Resonance, Doppler shift of solar radiation, Geocentric satellite, Amplitude and Time period.

1. INTRODUCTION

   In presence of different perturbation, the three-body problem and restricted three-body problem have been studied by different authors of the first half of the twentieth century. The perturbation due to mechanism of dissipation in solar system is multidimensional. The non-gravitational dissipative force due to gas nebula moving in solar space is known as Stokes drag which is proportional to the velocity of particle with respect to gas and valid for low Reynold number Re <10. The particular case of Stokes drag is the linear drag.

   To study solar dynamics, one should know about the factors influencing the motion of the bodies of the solar system. Resonance existing in solar system plays an important role in the solar dynamics. During the integration of equations of motion; a set of cases in which the periods of revolution are in the ratio of two integers manifested by the appearance of small divisors is described as resonance. Hughes (1980) studied the effect of resonance on the orbit of the Earths satellite due to lunisolar gravity and the corresponding direct solar radiation pressure, occurrence of which depends only on the satellites orbital inclination. Weidenchilling and Devis (1985) discovered the behaviour of resonance trapping in a gasrich scenario which is extended by Patterson (1987) for the existence of resonances of any order and showing formation of planetary embryous at two-body external resonances by accretion of infinitesimals caught in these orbits. Bhatnagar and Mehra (1986) examined the motion of a satellite by taking gravitational forces of the Moon, Earth and the radiating Sun. Ferraz-Mello (1992) studied averaging of the elliptic asteroidal problem with a Stokes drag and with the assistance of Beauge (1993) he studied resonance trapping and Stokes drag dissipation in the primordial solar nebula. The often decrease of semi-major axis due to dissipation and consequent collision between one primary and minor bodies has been studied by Celletti and Stefanelli (2011). Quasles et al. (2012) has studied the resonances for co-planar CR3BP for the mass ratio between 0.10 and 0.15 and used the method of maximum Lyaponav exponent to locate the resonant points. They showed that in presence of single resonance, the orbital stability is ensured for high value of resonance.

   Sushil et al. (2013) worked on resonance in a geocentric satellite due to Earths equatorial ellipticity and analysed the effects on amplitude and time period of oscillation on (angle measured from the

   minor axis of the Earths equatorial ellipse to the projection of the moon on the plane of equator) and on the other orbital elements of the satellite. Rosemary (2013) has given detail description of the perturbation theory to determine the presence of resonance based on approximations to a harmonic oscillation. Charanpreet Kaur et al. (2018) worked on resonance in the motion of geocentric satellite

   due to PR-drag and further in (2019) they have discussed the Resonance in the motion of geocentric satellite due to PR-drag and equatorial ellipticity of the Earth. Hassan et al (2022) studied effects of Stokes drag on the resonant motion of a geocentric satellite and found that time period and amplitude vary with the variation of Stokes Drag parameter. Presently we proposed to extend the work of Hassan et al (2022) by considering the Combined Effect of Stokes Drag and Earths equatorial ellipticity on the resonant motion of moon, where the minor axis of the Earths equatorial section is called ellipticity parameter of the Earth. Here the Stokes-Drag defined by Ferraz-Mello (1992) is under consideration.

   We divide this paper in five sections. In section 2, the equations of motion of the geocentric satellite in polar form have been established in presence of Doppler shift of solar radiation in rotating frame relative to the Earth. In Section 3, we have solved first the unperturbed equation of motion and hence the integrable form of the perturbed equation of motion and its solution is established. In section 4, Amplitudes and time periods have been found out by using the generalised formula of Hassan et. al (2022). The manuscript has been concluded in section 5 and ended with the references.

2. THE EQUATIONS OF MOTION

   Let us considering the inertial frame E, X 0Y0 Z

   frame E, XYZ relative to the inertial one , where

   whose origin at the Earth E and a rotating

   EX passes through the vernal equinox

   .Let i , j and i, j be the unit vectors along the axes of inertial frame and rotating frame with

   common unit vector k along the vertical axis EZ (not seen in the figure). Let EP r be the

   position vector of the satellite P, SP be the position of the Sun S relative to the Earth E and

   SE R . If M , m and be the masses of the Sun, Earth and the Geocentric Satellite

   respectively then their mutual gravitational forces are given by

   The Doppler shift of solar radiation applied on the satellite P is given by

   where 0,1 is the dissipative constant [(Beauge and Ferraz-Mello (1993)]. Let be

   angular velocity of the rotating frame relative to the inertial frame and the unit vector i along the direction of geocentric satellite then the equation of motion of satellite in rotating frame can be written as

   Let be the angle of direction of the satellite with the direction of vernal equinox, then k

   , where is the angular velocity of the satellite. Thus, the equation (3) reduced to

   In the triangle EPS

   r R FSP FEP DR FSE GM Gm r GM R.

   If R be the unit vector along R and be the angle of the direction of the sun with the direction

   of vernal equinox then R cos i sin j and 2 GM implies that

   Scalar product of i with (4) and (5) and that of j with 4 and 5 comparing the results one can find the equations of motions of the satellite in polar form as

   These equations are not integrable, so we replace r and by their steady state value r0 and 0/p>

   by perturbation technique which can be introduced in 6 and 7 as 0t & t

   For central orbit of satellite r 2 constant=h (say) and r 1/ u in (8) we get

3. RESONANCE IN THE MOTION OF THE SATELLITE

   The complete solution of the unperturbed equation of motion

   Let us consider 0t nt(say) where n be the frequency of the satellite. Since eccentricity e 1 , so (1 e cos nt)p 1 h e cos nt .

   d 2u 2 d 2u

   Hence by using n and in equation (10), then we get the perturbed equation of

   dt2 0 d 2

   the motion of the satellite is

   The solution of equation (11) is given by

   Where is constant of integration. On vanishing the denominator of any term of equation (12) we get some points at which motion becomes indeterminate and hence resonance occurs at these points. Thus, the resonances occur at the points n , n 2 , n 3 , 2n

   2n 3 , 3n ,3n 2 , 4n , 4n 3 , 5n and 5n 3 . All the resonances 1:1, 1: 2, 1: 3, 2 :1 2 : 3, 3 :1, 3 : 2, 4 :1, 4 : 3, 5 :1 and 5 : 3 occur due to doppler shift of solar radiation.

4. AMPLITUDE AND TIME PERIOD

   By using Brown and shook (1933) and Hassan et.al (2022) the generalization formula of the

   amplitude A and the time period T at the resonant point m1n m2 where m1, m2 N for

   the equation is of the form

   Its to be noted that any value of s may or may not represent the corresponding value of A and T

   using the result of 13 the amplitude and time period at different resonant points are cited in the table.

   Resonant Point	Amplitude	Time Period	s
   n	A1 A2 A3 A4 A5 A6	T1 T2 T3 T4 T5 T6	5 13 16 20 22 25
   n 2	A7 A8	T7 T8	6 19
   n 3	A9	T9	7
   2n	A10 A11 A12	T10 T11 T12	11 21 22
   2n 3	A13	T13	15
   3n	A14	T14	16
   3n 2	A15	T15	19
   4n	A16	T16	21
   4n 3	A17	T17	23
   5n	A18	T18	24
   5n 3	A19	T19	25

   here

5. CONCLUSION

In section 1, of this manuscript, the previous works have been cited. In section 2, the polar equations of motion of the geocentric satellite have been established in presence of doppler shift of solar radiation in rotating frame relative to the Earth. To reduce the chances of non-integrability of the equations of motion, we used perturbation technique by taking the steady state values of the position vector and angular velocity of satellite. In section 3, we have solved first the unperturbed equation of motion. The solution of perturbed equation (11) of motion in equation (12). By making denominator of any term from 5th to 26th to zero u becomes infinity and hence the motion of the satellite becomes indeterminate. Thus n , n 2 , n 3 , 2n , 2n 3 , 3n , 3n 2 ,

4n , 4n 3 , 5n and 5n 3 are eleven resonances of the problem all of them are

occurred due to Doppler shift of solar radiation. In section 4, we have found the amplitudes and time periods at all the resonant points which are occurred due to Doppler shift of solar radiation.

REFERENCES

1. A. Celletti, L. Stefanelli, E. Lega, C. Froeschle, Some results on the global dynamics of the regularized restricted three-body problem with dissipation. CelestMechDynAstr 109, 265284 (2011).
2. E.W. Brown, C.A. Shook, Planetary Theory (Cambridge at the University Press, Cambridge,
   1. (1933).
3. D. Brouwer. Analytical study of resonance caused by solar radiation pressure, in Dynamics of Satellite Symposium Paris, pp. 2830, (1963).
4. K.B. Bhatnagar, M. Mehera, The motion of a geosynchronous satellite-I. Indian J. Pure Appl. Math. 17(12), 14381452 (1986).
5. C. Beauge, S. Ferraz-Mello, Resonance trapping in the primordial solar nebula: the case of a stokes drag dissipation. Icarus 103, 301 318 (1993).
6. C. Kour, B.K. Sharma, L.P. Pandey, Resonance in the motion of geocentric satellite due to Poynting-Robertson drag. An Int. J. 13, 173 189 (2018).
7. C. Kour, B.K. Sharma, S. Yadav, Resonance in the motion of a geocentric satellite due to Poynting-Robertson drag and Equatorial Ellipticity of the Earth. An Int. J. 14, 416436 (2019).
8. S. Hughes, Earth Satellite orbits with resonant lunisolar perturbations. I. Resonances dependent only on inclination. Proc. R. Soc. Lond. Ser. A372, 243264 (1980).
9. C.D. Murray, S.F. Dermott, Solar System Dynamics (Cambridge University Press, Cambridge, (2010).
10. M.R. Hassan, Md Sabir Ahamad, B.K. Sharma, The Effects of Stokes drag on the Resonant Motion of a Geocentric Satellite. Few-Body Syst 63:76 (2022).
11. C.W. Patterson, Resonance capture and the evolution of the planets. Icarus 70, 319333 (1987).
12. J.H. Poynting, Radiation in the solar system, its effect on temperature and its pressure on small bodies. Philos. Trans. R. Soc. London A.

    202, 525552 (1903).

13. B. Quarles, Z.E. Musielak, M. Cuntz, Study of resonances for the restricted 3-body problem. Asron. Nachr. 00, 110 (2012).
14. H. Robertson, Dynamical effects of radiation in the Solar System. Mon. Not. R. Astron. Soc. 97, 423437 (1937).
15. S.J. Weidenschilling, D.R. Davis, Orbital resonances in the solar nebula. Implications for planetary accretion. ICARUS 62, 1629 (1985).
16. S. Yadav, R. Aggarwal, Resonance in a geo-centric satellite due to Earths equatorial ellipticity. Astrophys. Space Sci. 347, 249259 (2013).