In this work we study the perturbation and the change in the orbital elements due to the earth’s magnetic field and the gravitational waves. The acceleration components are derived in the radial, transverse to it and...In this work we study the perturbation and the change in the orbital elements due to the earth’s magnetic field and the gravitational waves. The acceleration components are derived in the radial, transverse to it and normal to the orbital plane. The equation for the rates of variation of the elements is formed and solved to find the secular variation in the element for polar and equatorial satellites.展开更多
The optimizing total velocity increment Δv needed for orbital maneuver between two elliptic orbits with plane change is investigated. Two-impulse orbital transfer is used based on a changing of transfer velo...The optimizing total velocity increment Δv needed for orbital maneuver between two elliptic orbits with plane change is investigated. Two-impulse orbital transfer is used based on a changing of transfer velocities concept due to the changing in the energy. The transferring has been made between two elliptic orbits having a common centre of attraction with changing in their planes in standard Hohmann transfer with the terminal orbit which is elliptic orbit and not circular. We develop a treatment based on the elements of elliptic orbits a1,e1, a2,e2, and?aT,eT of the initial orbit, final orbit and transferred orbit respectively. The first impulse Δv1 at the perigee induces a rotation of the orbital plane by ?which will be minimized. The second impulse Δv2 at apogee is induced an angle ?to product the final elliptic orbit. The total plane change required . We calculate the total impulse Δv and minimize by optimizing angle of plane’s variation . We obtain a polynomial equation of six degrees on the two transfer angles between neither two elliptic orbits ?and . The solution obtained numerically, using programming code of MATHEMATICA V10, with no condition on the eccentricity or the semi-major axis of the initial, transformed, and the final orbits. We find that there are constrains on the transfer angles and α. For αit must be between 40°and 160°, and there is no solution if αis less than 40°and bigger than 160°and ?takes the values less than 40°. The minimum total velocity increments obtained at the value of ?less than 25°and& alpha;equal to 160°. This is an interesting result in orbital transfer problem in which the change of orbital plane is necessary for the transferring.展开更多
We compute the long-term orbital variation of a test particle orbiting a central body acted upon by normal incident of plane gravitational wave. We use the tools of celestial mechanics to give the first order solution...We compute the long-term orbital variation of a test particle orbiting a central body acted upon by normal incident of plane gravitational wave. We use the tools of celestial mechanics to give the first order solution of canonical equations of long-period and short-period terms of the perturbed Hamiltonian of gravitational waves. We consider normal incident of plane gravitational wave and characteristic size of bound—two body system (earth’s satellite or planet) is much smaller than the wavelength of the wave and the wave’s frequency nw is much smaller than the particle’s orbital np. We construct the Hamiltonian of the gravitational waves in terms of the canonical variables (l,g,h,L,G,H)?and we solve the canonical equations numerically using Runge-Kutta fourth order method using language MATHEMATICA V10. Taking Jupiter as practical example we found that there are long period perturbations on ω,Ωand i?and not changing with revolution and the short period perturbations on a, e and M?changing with revolution during the interval of time (t−t0 ) which is changing from 0→4π.展开更多
This work deals with the numerical solution of the gravitational waves effects on the orbital elements of the planets in case of commensurability between the wave’s frequency ng and the planet’s mean motion np. Taki...This work deals with the numerical solution of the gravitational waves effects on the orbital elements of the planets in case of commensurability between the wave’s frequency ng and the planet’s mean motion np. Taking Mercury and Pluto as practical examples for low frequency and high frequency, the variations of the orbital elements of Mercury due to resonance of gravitational wave are different and small than the perturbation on Pluto. The amount of changing in the orbital elements under the effects of gravitational waves is different from planet to planet according to the planet’s mean motion np. For low frequency ng, the secular variation in orbital elements will be negative (i.e. decreasing) in the inclination, semi-major axis and the eccentricity (i, a, e) like as Pluto. For high frequency ng like Mercury, the secular variation in all the orbital elements will be positive (i.e. increasing). The perturbation on all the orbital elements of two planets is changing during each revolution except the eccentricity e of Mercury and the mean anomaly M of Mercury and Pluto during the time.展开更多
For certain values of semi-major axis and eccentricity, orbit plane precession caused by Earth oblate is synchronous with the mean orbital motion of the apparent Sun (a sun-synchronism). Many forces cause slow changes...For certain values of semi-major axis and eccentricity, orbit plane precession caused by Earth oblate is synchronous with the mean orbital motion of the apparent Sun (a sun-synchronism). Many forces cause slow changes in the inclination and ascending node of sun-synchronous orbits. In this work, we investigate the analytical perturbations due to the direct solar radiation pressure and gravitational waves effects. A full analytical solution is obtained using technique of canonical Lie-transformation up to the order three in (the oblateness of the Earth). The solar radiation pressure and gravitational waves perturbations cause second order effects on all the elements of the elliptic orbit (the eccentricity, inclination, ascending node, argument of perigee, and semi-major axis) consequently these perturbations will cause disturbance in the sun-synchronism. Also we found that the perturbation or the behavior of gravitational waves almost the same as the perturbation or the behavior of solar radiation pressure and their coupling will incorporate the sun-synchronism through the secular rate of the ascending node precession.展开更多
文摘In this work we study the perturbation and the change in the orbital elements due to the earth’s magnetic field and the gravitational waves. The acceleration components are derived in the radial, transverse to it and normal to the orbital plane. The equation for the rates of variation of the elements is formed and solved to find the secular variation in the element for polar and equatorial satellites.
文摘The optimizing total velocity increment Δv needed for orbital maneuver between two elliptic orbits with plane change is investigated. Two-impulse orbital transfer is used based on a changing of transfer velocities concept due to the changing in the energy. The transferring has been made between two elliptic orbits having a common centre of attraction with changing in their planes in standard Hohmann transfer with the terminal orbit which is elliptic orbit and not circular. We develop a treatment based on the elements of elliptic orbits a1,e1, a2,e2, and?aT,eT of the initial orbit, final orbit and transferred orbit respectively. The first impulse Δv1 at the perigee induces a rotation of the orbital plane by ?which will be minimized. The second impulse Δv2 at apogee is induced an angle ?to product the final elliptic orbit. The total plane change required . We calculate the total impulse Δv and minimize by optimizing angle of plane’s variation . We obtain a polynomial equation of six degrees on the two transfer angles between neither two elliptic orbits ?and . The solution obtained numerically, using programming code of MATHEMATICA V10, with no condition on the eccentricity or the semi-major axis of the initial, transformed, and the final orbits. We find that there are constrains on the transfer angles and α. For αit must be between 40°and 160°, and there is no solution if αis less than 40°and bigger than 160°and ?takes the values less than 40°. The minimum total velocity increments obtained at the value of ?less than 25°and& alpha;equal to 160°. This is an interesting result in orbital transfer problem in which the change of orbital plane is necessary for the transferring.
文摘We compute the long-term orbital variation of a test particle orbiting a central body acted upon by normal incident of plane gravitational wave. We use the tools of celestial mechanics to give the first order solution of canonical equations of long-period and short-period terms of the perturbed Hamiltonian of gravitational waves. We consider normal incident of plane gravitational wave and characteristic size of bound—two body system (earth’s satellite or planet) is much smaller than the wavelength of the wave and the wave’s frequency nw is much smaller than the particle’s orbital np. We construct the Hamiltonian of the gravitational waves in terms of the canonical variables (l,g,h,L,G,H)?and we solve the canonical equations numerically using Runge-Kutta fourth order method using language MATHEMATICA V10. Taking Jupiter as practical example we found that there are long period perturbations on ω,Ωand i?and not changing with revolution and the short period perturbations on a, e and M?changing with revolution during the interval of time (t−t0 ) which is changing from 0→4π.
文摘This work deals with the numerical solution of the gravitational waves effects on the orbital elements of the planets in case of commensurability between the wave’s frequency ng and the planet’s mean motion np. Taking Mercury and Pluto as practical examples for low frequency and high frequency, the variations of the orbital elements of Mercury due to resonance of gravitational wave are different and small than the perturbation on Pluto. The amount of changing in the orbital elements under the effects of gravitational waves is different from planet to planet according to the planet’s mean motion np. For low frequency ng, the secular variation in orbital elements will be negative (i.e. decreasing) in the inclination, semi-major axis and the eccentricity (i, a, e) like as Pluto. For high frequency ng like Mercury, the secular variation in all the orbital elements will be positive (i.e. increasing). The perturbation on all the orbital elements of two planets is changing during each revolution except the eccentricity e of Mercury and the mean anomaly M of Mercury and Pluto during the time.
文摘For certain values of semi-major axis and eccentricity, orbit plane precession caused by Earth oblate is synchronous with the mean orbital motion of the apparent Sun (a sun-synchronism). Many forces cause slow changes in the inclination and ascending node of sun-synchronous orbits. In this work, we investigate the analytical perturbations due to the direct solar radiation pressure and gravitational waves effects. A full analytical solution is obtained using technique of canonical Lie-transformation up to the order three in (the oblateness of the Earth). The solar radiation pressure and gravitational waves perturbations cause second order effects on all the elements of the elliptic orbit (the eccentricity, inclination, ascending node, argument of perigee, and semi-major axis) consequently these perturbations will cause disturbance in the sun-synchronism. Also we found that the perturbation or the behavior of gravitational waves almost the same as the perturbation or the behavior of solar radiation pressure and their coupling will incorporate the sun-synchronism through the secular rate of the ascending node precession.
