A study on the orbital lifetime estimation of upper stages with different inclinations is carried... more A study on the orbital lifetime estimation of upper stages with different inclinations is carried out through response surface technique. The combined influence of the lunisolar perturbations and drag on lifetime variations plays an important role. In order to extract information on the lifetime from orbital data, the estimation of area to mass ratio, initial perigee altitude and ballistic coefficient is performed. In this context, it is noticed that when an object undergoes orbital resonance, bifurcation between the observed and predicted trajectories takes place. However, in some scenarios, simulation of apogee and perigee altitude profiles can still be performed to estimate relevant parameters. To capture only secular changes in apogee and perigee altitudes, osculating elements can be converted to their respective mean values by utilizing a suitable theory. Furthermore, a study on the re-entry of the object is examined closely by estimating different ballistic coefficients to match the position at different epochs. The non-uniform change in the ballistic coefficient provides an indication of possible chaotic motion.
The collinear Lagrange points of the Earth-Moon system provide an ideal environment for future mi... more The collinear Lagrange points of the Earth-Moon system provide an ideal environment for future missions. L 1 point, which lies between the Earth and the Moon, has potential for a manned space station to transport cargo and personnel to the Moon and back. Similarly, L 2 point can be a candidate location for communication satellites covering the far side of Moon. Because, Lagrange Points promise to be the hub of future space operations, it has become important to study effect of a spacecraft fragmentation at these points. In this context, Stumpff/Weiss four-body algorithm, which is an extension of the Encke method of orbit propagation, provides a very attractive proposition for the simulation of fragment evolution. The method is 10 to 15 times faster than the other similar techniques and hence permits Monte-Carlo (MC) analysis of fragmentation velocity. Following a fragmentation at Earth-Moon collinear point about 2% of the total number of debris pieces can come within GSO altitude (~ 3.6×10 4 km). Fragmentation at any one of the Earth-Moon collinear points poses small yet perceptible risk to space operation around the Earth. It is emphasized that there is a genuine need to conduct more detailed study on fragmentation at collinear Earth-Moon points.
International Journal of Astronomy and Astrophysics, 2016
The Circular Restricted Three-Body Problem (CRTBP) with more massive primary as an oblate spheroi... more The Circular Restricted Three-Body Problem (CRTBP) with more massive primary as an oblate spheroid with its equatorial plane coincident with the plane of motion of the primaries is considered to generate the halo orbits around L1 and L2 for the seven satellites (Mimas, Enceladus, Tethys, Dione, Rhea, Titan and Iapetus) of Saturn in the frame work of CRTBP. It is found that the oblateness effect of Saturn on the halo orbits of the satellites closer to Saturn has significant effect compared to the satellites away from it. The halo orbits L1 and L2 are found to move towards Saturn with oblateness.
International Journal of Astronomy and Astrophysics, 2016
This paper deals with generation of halo orbits in the three-dimensional photogravitational restr... more This paper deals with generation of halo orbits in the three-dimensional photogravitational restricted three-body problem, where the more massive primary is considered as the source of radiation and the smaller primary is an oblate spheroid with its equatorial plane coincident with the plane of motion. Both the terms due to oblateness of the smaller primary are considered. Numerical as well as analytical solutions are obtained around the Lagrangian point L1, which lies between the primaries, of the Sun-Earth system. A comparison with the real time flight data of SOHO mission is made. Inclusion of oblateness of the smaller primary can improve the accuracy. Due to the effect of radiation pressure and oblateness, the size and the orbital period of the halo orbit around L1 are found to increase.
International Journal of Astronomy and Astrophysics, 2016
We analyze the periodic orbits, quasi periodic orbits and chaotic orbits in the photo gravitation... more We analyze the periodic orbits, quasi periodic orbits and chaotic orbits in the photo gravitational Sun-Saturn system incorporating actual oblateness of Saturn in the planar circular restricted three body problem. In this paper, we study the effect of solar radiation pressure on the location of Sun centered and Saturn centered orbits, its diameter, semi major axis and eccentricity by taking different values of solar radiation pressure q and different values of Jacobi constant "C", and by considering actual oblateness of Saturn using Poincare surface of section (PSS) method. It is observed that by the introduction of perturbing force due to solar radiation pressure admissible range of Jacobi constant C decreases, it is also observed that as value of C decreases the number of islands decreases and as a result the number of periodic and quasi periodic orbits decreases. Further, the periodic orbits around Saturn and Sun moves towards Sun by decreasing perturbation due to solar radiation pressure q for a specific choice of Jacobi constant C. It is also observed that due to solar radiation pressure, semi major axis and eccentricity of Sun centered periodic orbit reduces, whereas, due to solar radiation pressure uniform change in semi major axis and eccentricity of Saturn centered periodic orbits is observed.
International Journal of Astronomy and Astrophysics, 2019
We study halo orbits in the circular restricted three-body problem (CRTBP) with both the primarie... more We study halo orbits in the circular restricted three-body problem (CRTBP) with both the primaries as the sources of radiation. The positioning of the triangular equilibrium points is discussed in a rotating coordinate system.
A new non-singular, analytical theory with respect to the Earth's zonal harmonic term J 2 has bee... more A new non-singular, analytical theory with respect to the Earth's zonal harmonic term J 2 has been developed for short-periodic motion, by analytically integrating the uniformly regular KS canonical equations of motion using generalized eccentric anomaly 'E' as the independent variable. Only one of the eight equations needs to be integrated analytically to generate the state vector, as a result of symmetry in the equations of motion, and the computation for the other equations is by changing the initial conditions. The integrals are much simpler than earlier obtained in [20] in terms of the independent variable 's'. Numerical results indicate that the solution is reasonably accurate for a wide range of orbital parameters during a revolution. The error in computing the most important orbital parameter 'semi-major axis' which is the measure of energy is less than five percentage during a revolution. The analytical solution can have number of applications. It can be used for studying the short-term relative motion of two or more space objects. It can also be useful in collision avoidance studies of space objects. It can be used for onboard computation in the navigation and guidance packages, where the modeling of J 2 effect becomes necessary.
Photogravitational Restricted Three-Body Problem (PRTBP) with smaller primary being an oblate sph... more Photogravitational Restricted Three-Body Problem (PRTBP) with smaller primary being an oblate spheroid with its equatorial plane coincident with the plane of motion of the primaries is considered and halo orbits in the vicinity of Sun-Mars Lagrangian points L1 and L2 are computed numerically. The effects of perturbations on size, shape, location and time period of the halo orbits are studied. It is found that the increase in solar radiation pressure at constant oblateness elongates the halo orbits at L1 and the orbits move towards the radiating body. At L2, the halo orbits shrink and move towards the smaller primary with increase in solar radiation pressure at constant oblateness. For constant radiation pressure, increase in oblateness causes the location of L1 and L2 halo orbits to move away from the smaller primary. The time period of L1 halo orbits increases with increase in radiation pressure for constant oblateness and decreases with increase in oblateness for constant radiation pressure. However, the effect of solar radiation pressure and oblateness for L2 halo orbits is reversed.
Sharma’s singularity-free analytical theory for the short-term orbital motion of satellites in te... more Sharma’s singularity-free analytical theory for the short-term orbital motion of satellites in terms of KS elements in closed form in eccentricity with Earth’s zonal harmonic term J2, is improved by using King-Hele’s expression for the radial distance ‘r’ which includes the effect of J2, and is suitable for low eccentricity orbits. Numerical experimentation with four test cases with perigee altitude of 200 km and eccentricity varying from 0.01 to 0.3 for different inclinations is carried out. It is found that the orbital elements computed with the analytical expressions in a single step during half a revolution match very well with numerically integrated values and show significant improvement over the earlier theory. The solution can be effectively used for computation of mean elements for near-Earth orbits, where the short-term orbit perturbations due to J2 play most important role. The theory will be very useful in computing the state vectors during the coast phase of rocket traj...
Background/Objectives: This study deals wit h the stationary solutions of the planar circular res... more Background/Objectives: This study deals wit h the stationary solutions of the planar circular restricted three-body problem when the more massive primary is a source of radiation and the smaller primary is an oblate spheroid with its equatorial plane coincident with the plane of motion. The objective is to study the location of the Lagrangian points and to find the values of critical mass. Also, to study the periodic orbits around the Lagrangian points. Methods: A new mean motion expression by including the secular perturbation due to oblateness utilized by (1,2) is used in the present studies. The characteristic roots are obtained by linearizing the equation of the motion around the Lagrangian points. Findings: The critical mass parameter µcrit (3,4) , which decreases radiation force, whereas it increases with oblateness when we consider the value of new mean motion. Through special choice of initial conditions, retrograde elliptical periodic orbits exist for the case µ = µcrit, whose eccentricity increases with oblateness and decreases with radiation force for non-zero oblateness, although there is slight variation in L 2 location.
Background: The location and stability of the equilibrium points are studied for the Planar Circu... more Background: The location and stability of the equilibrium points are studied for the Planar Circular Restricted Three-Body Problem where the more massive primary is an oblate spheroid. Methods: The mean motion of the equations of motion is formulated from the secular perturbations as derived by(1) and used in(2–4). The singularities of the equations of motion are found for locating the equilibrium points. Their stability is analysed using the linearized variational equations of motion at the equilibrium points. Findings: As the effect of oblateness in the mean motion expression increases, the location and stability of the equilibrium points are affected by the oblateness of the more massive primary. It is interesting to note that all the three collinear points move towards the more massive primary with oblateness. It is a new result. Among the shifts in the locations of the five equilibrium points, the y–location of the triangular equilibrium points relocate the most. It is very int...
Heat transfer in mixed convection unsteady MHD flow of an incompressible nanofluid in a channel u... more Heat transfer in mixed convection unsteady MHD flow of an incompressible nanofluid in a channel under the influence of heat source is studied. The channel with non-uniform walls temperature is taken in a perpendicular direction with a transverse magnetic field. Based on the substantial boundary conditions, three different flow conditions are examined. The problem is formed in PDEs with substantial boundary conditions. Four different forms of nanoparticles of identical volume fraction are employed in traditional base fluid water (H 2 O). Solutions for momentum and energy are attained by the perturbation method and examined graphically in different graphs. It is established that viscosity and thermal conductivity are the mainly well-known variables accountable for different results of velocity and temperature. It is also found that increasing heat source leads to an increase in nanofluid velocity and temperature and nano-size particles instance platelet and blade shapes have lesser mo...
The triangular Lagrangian points of the elliptic restricted three-body problem (ERTBP) with oblat... more The triangular Lagrangian points of the elliptic restricted three-body problem (ERTBP) with oblate and radiating more massive primary are studied. The mean motion equation used here is different from the ones employed in many studies on the perturbed ERTBP. The effect of oblateness on the mean motion equation varies. This change influences the location and stability of the triangular Lagrangian points. The points tend to shift in the y-direction. The influence of the oblateness on the critical mass ratio is also altered. But the eccentricity limit for stability remains the same.
International Journal of Astronomy and Astrophysics, 2011
An analytical theory for calculating perturbations of the orbital elements of a satellite due to ... more An analytical theory for calculating perturbations of the orbital elements of a satellite due to J 2 to accuracy up to fourth power in eccentricity is developed. It is observed that there is significant improvement in all the orbital elements with the present theory over second-order theory. The theory is used for computing the mean orbital elements, which are found to be more accurate than the existing Bhatnagar and Taqvi's theory (up to second power in eccentricity). Mean elements have a large number of practical applications.
The combination of atmospheric drag and lunar and solar perturbations in addition to Earth’s obla... more The combination of atmospheric drag and lunar and solar perturbations in addition to Earth’s oblateness influences the orbital lifetime of an upper stage in geostationary transfer orbit (GTO). These high eccentric orbits undergo fluctuations in both perturbations and velocity and are very sensitive to the initial conditions. The main objective of this paper is to predict the reentry time of the upper stage of the Indian geosynchronous satellite launch vehicle, GSLV-D5, which inserted the satellite GSAT-14 into a GTO on January 05, 2014, with mean perigee and apogee altitudes of 170 km and 35975 km. Four intervals of near linear variation of the mean apogee altitude observed were used in predicting the orbital lifetime. For these four intervals, optimal values of the initial osculating eccentricity and ballistic coefficient for matching the mean apogee altitudes were estimated with the response surface methodology using a genetic algorithm. It was found that the orbital lifetime from...
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1992
Analytical theory for the motion of near-Earth satellite orbits with the air drag effect is devel... more Analytical theory for the motion of near-Earth satellite orbits with the air drag effect is developed in terms of the KS elements, utilizing an analytical oblate exponential atmospheric model. The series expansions include up to cubic terms in e (eccentricity) and c (a small parameter dependent on the flattening of the atmosphere). Due to the symmetry of the KS element equations, only one of the eight equations is integrated analytically to obtain the state vector at the end of each revolution. Numerical comparisons are made with nine test cases, selected to cover a wide range of eccentricity with perigee heights near to 300 km at three different inclinations. A comparison of three orbital parameters: semi-major axis, eccentricity and argument of perigee, perturbed by air drag with oblate atmosphere is made with the previously developed second-order theory. It is found that with the present theory with increase in eccentricity there is improvement in semi-major axis and eccentricity...
Poincaré surface of section technique is used to study the evolution of a family 'f ' of simply s... more Poincaré surface of section technique is used to study the evolution of a family 'f ' of simply symmetric retrograde periodic orbits around the smaller primary in the framework of restricted three-body problem for a number of systems, actual and hypothetical, with mass ratio varying from 10 −7 to 0.015. It is found that as the mass ratio decreases the region of phase space containing the two separatrices shrinks in size and moves closer to the smaller primary. Also the corresponding value of Jacobi constant tends towards 3.
It is well known (cf. Szebehely, 1967a) that, in the restricted three-body problem, the condition... more It is well known (cf. Szebehely, 1967a) that, in the restricted three-body problem, the condition for linear stability of the two equidistant points of equilibrium is that the mass parameter # is less than the critical value
The motion of a particle in the restricted three-body problem is explored by treating the more ma... more The motion of a particle in the restricted three-body problem is explored by treating the more massive primary as an oblate spheroid with its equatorial plane coincident with the plane of motion of the primaries using a perturbation method. Initial conditions for the infinitesimal periodic orbits around the more massive primary are generated and the effect of oblateness on the perigee of these orbits is studied as well. It is observed that when oblateness coefficient is increased, the perigee of the orbit shifts towards both the primaries depending upon the increase in period and mass ratio. It is further noticed that during this transition, for certain periods, the perigee of the orbit remains unaltered with the increase in the oblateness coefficient.
A new non-singular analytical theory with respect to the Earth’s zonal harmonic terms J2, J3, J4 ... more A new non-singular analytical theory with respect to the Earth’s zonal harmonic terms J2, J3, J4 has been developed for short-periodic motion, by analytically integrating the uniformly regular KS canonical equations of motion using generalized eccentric anomaly ‘E’ as the independent variable. Only one of the eight equations need to be integrated analytically to generate the state vector, due to the symmetry in the equations of motion, and the computation for the other equations is done by changing the initial conditions. King-Hele’s expression for radial distance ‘r’ with J2 is also considered in generating the solution. The results obtained from the analytical expressions in a single step during half a revolution match quite well with numerically integrated values. Numerical results also indicate that the solution is reasonably accurate for a wide range of orbital elements during half a revolution and is an improvement over Sharon et al. [17] theory, which is generated in terms of...
A study on the orbital lifetime estimation of upper stages with different inclinations is carried... more A study on the orbital lifetime estimation of upper stages with different inclinations is carried out through response surface technique. The combined influence of the lunisolar perturbations and drag on lifetime variations plays an important role. In order to extract information on the lifetime from orbital data, the estimation of area to mass ratio, initial perigee altitude and ballistic coefficient is performed. In this context, it is noticed that when an object undergoes orbital resonance, bifurcation between the observed and predicted trajectories takes place. However, in some scenarios, simulation of apogee and perigee altitude profiles can still be performed to estimate relevant parameters. To capture only secular changes in apogee and perigee altitudes, osculating elements can be converted to their respective mean values by utilizing a suitable theory. Furthermore, a study on the re-entry of the object is examined closely by estimating different ballistic coefficients to match the position at different epochs. The non-uniform change in the ballistic coefficient provides an indication of possible chaotic motion.
The collinear Lagrange points of the Earth-Moon system provide an ideal environment for future mi... more The collinear Lagrange points of the Earth-Moon system provide an ideal environment for future missions. L 1 point, which lies between the Earth and the Moon, has potential for a manned space station to transport cargo and personnel to the Moon and back. Similarly, L 2 point can be a candidate location for communication satellites covering the far side of Moon. Because, Lagrange Points promise to be the hub of future space operations, it has become important to study effect of a spacecraft fragmentation at these points. In this context, Stumpff/Weiss four-body algorithm, which is an extension of the Encke method of orbit propagation, provides a very attractive proposition for the simulation of fragment evolution. The method is 10 to 15 times faster than the other similar techniques and hence permits Monte-Carlo (MC) analysis of fragmentation velocity. Following a fragmentation at Earth-Moon collinear point about 2% of the total number of debris pieces can come within GSO altitude (~ 3.6×10 4 km). Fragmentation at any one of the Earth-Moon collinear points poses small yet perceptible risk to space operation around the Earth. It is emphasized that there is a genuine need to conduct more detailed study on fragmentation at collinear Earth-Moon points.
International Journal of Astronomy and Astrophysics, 2016
The Circular Restricted Three-Body Problem (CRTBP) with more massive primary as an oblate spheroi... more The Circular Restricted Three-Body Problem (CRTBP) with more massive primary as an oblate spheroid with its equatorial plane coincident with the plane of motion of the primaries is considered to generate the halo orbits around L1 and L2 for the seven satellites (Mimas, Enceladus, Tethys, Dione, Rhea, Titan and Iapetus) of Saturn in the frame work of CRTBP. It is found that the oblateness effect of Saturn on the halo orbits of the satellites closer to Saturn has significant effect compared to the satellites away from it. The halo orbits L1 and L2 are found to move towards Saturn with oblateness.
International Journal of Astronomy and Astrophysics, 2016
This paper deals with generation of halo orbits in the three-dimensional photogravitational restr... more This paper deals with generation of halo orbits in the three-dimensional photogravitational restricted three-body problem, where the more massive primary is considered as the source of radiation and the smaller primary is an oblate spheroid with its equatorial plane coincident with the plane of motion. Both the terms due to oblateness of the smaller primary are considered. Numerical as well as analytical solutions are obtained around the Lagrangian point L1, which lies between the primaries, of the Sun-Earth system. A comparison with the real time flight data of SOHO mission is made. Inclusion of oblateness of the smaller primary can improve the accuracy. Due to the effect of radiation pressure and oblateness, the size and the orbital period of the halo orbit around L1 are found to increase.
International Journal of Astronomy and Astrophysics, 2016
We analyze the periodic orbits, quasi periodic orbits and chaotic orbits in the photo gravitation... more We analyze the periodic orbits, quasi periodic orbits and chaotic orbits in the photo gravitational Sun-Saturn system incorporating actual oblateness of Saturn in the planar circular restricted three body problem. In this paper, we study the effect of solar radiation pressure on the location of Sun centered and Saturn centered orbits, its diameter, semi major axis and eccentricity by taking different values of solar radiation pressure q and different values of Jacobi constant "C", and by considering actual oblateness of Saturn using Poincare surface of section (PSS) method. It is observed that by the introduction of perturbing force due to solar radiation pressure admissible range of Jacobi constant C decreases, it is also observed that as value of C decreases the number of islands decreases and as a result the number of periodic and quasi periodic orbits decreases. Further, the periodic orbits around Saturn and Sun moves towards Sun by decreasing perturbation due to solar radiation pressure q for a specific choice of Jacobi constant C. It is also observed that due to solar radiation pressure, semi major axis and eccentricity of Sun centered periodic orbit reduces, whereas, due to solar radiation pressure uniform change in semi major axis and eccentricity of Saturn centered periodic orbits is observed.
International Journal of Astronomy and Astrophysics, 2019
We study halo orbits in the circular restricted three-body problem (CRTBP) with both the primarie... more We study halo orbits in the circular restricted three-body problem (CRTBP) with both the primaries as the sources of radiation. The positioning of the triangular equilibrium points is discussed in a rotating coordinate system.
A new non-singular, analytical theory with respect to the Earth's zonal harmonic term J 2 has bee... more A new non-singular, analytical theory with respect to the Earth's zonal harmonic term J 2 has been developed for short-periodic motion, by analytically integrating the uniformly regular KS canonical equations of motion using generalized eccentric anomaly 'E' as the independent variable. Only one of the eight equations needs to be integrated analytically to generate the state vector, as a result of symmetry in the equations of motion, and the computation for the other equations is by changing the initial conditions. The integrals are much simpler than earlier obtained in [20] in terms of the independent variable 's'. Numerical results indicate that the solution is reasonably accurate for a wide range of orbital parameters during a revolution. The error in computing the most important orbital parameter 'semi-major axis' which is the measure of energy is less than five percentage during a revolution. The analytical solution can have number of applications. It can be used for studying the short-term relative motion of two or more space objects. It can also be useful in collision avoidance studies of space objects. It can be used for onboard computation in the navigation and guidance packages, where the modeling of J 2 effect becomes necessary.
Photogravitational Restricted Three-Body Problem (PRTBP) with smaller primary being an oblate sph... more Photogravitational Restricted Three-Body Problem (PRTBP) with smaller primary being an oblate spheroid with its equatorial plane coincident with the plane of motion of the primaries is considered and halo orbits in the vicinity of Sun-Mars Lagrangian points L1 and L2 are computed numerically. The effects of perturbations on size, shape, location and time period of the halo orbits are studied. It is found that the increase in solar radiation pressure at constant oblateness elongates the halo orbits at L1 and the orbits move towards the radiating body. At L2, the halo orbits shrink and move towards the smaller primary with increase in solar radiation pressure at constant oblateness. For constant radiation pressure, increase in oblateness causes the location of L1 and L2 halo orbits to move away from the smaller primary. The time period of L1 halo orbits increases with increase in radiation pressure for constant oblateness and decreases with increase in oblateness for constant radiation pressure. However, the effect of solar radiation pressure and oblateness for L2 halo orbits is reversed.
Sharma’s singularity-free analytical theory for the short-term orbital motion of satellites in te... more Sharma’s singularity-free analytical theory for the short-term orbital motion of satellites in terms of KS elements in closed form in eccentricity with Earth’s zonal harmonic term J2, is improved by using King-Hele’s expression for the radial distance ‘r’ which includes the effect of J2, and is suitable for low eccentricity orbits. Numerical experimentation with four test cases with perigee altitude of 200 km and eccentricity varying from 0.01 to 0.3 for different inclinations is carried out. It is found that the orbital elements computed with the analytical expressions in a single step during half a revolution match very well with numerically integrated values and show significant improvement over the earlier theory. The solution can be effectively used for computation of mean elements for near-Earth orbits, where the short-term orbit perturbations due to J2 play most important role. The theory will be very useful in computing the state vectors during the coast phase of rocket traj...
Background/Objectives: This study deals wit h the stationary solutions of the planar circular res... more Background/Objectives: This study deals wit h the stationary solutions of the planar circular restricted three-body problem when the more massive primary is a source of radiation and the smaller primary is an oblate spheroid with its equatorial plane coincident with the plane of motion. The objective is to study the location of the Lagrangian points and to find the values of critical mass. Also, to study the periodic orbits around the Lagrangian points. Methods: A new mean motion expression by including the secular perturbation due to oblateness utilized by (1,2) is used in the present studies. The characteristic roots are obtained by linearizing the equation of the motion around the Lagrangian points. Findings: The critical mass parameter µcrit (3,4) , which decreases radiation force, whereas it increases with oblateness when we consider the value of new mean motion. Through special choice of initial conditions, retrograde elliptical periodic orbits exist for the case µ = µcrit, whose eccentricity increases with oblateness and decreases with radiation force for non-zero oblateness, although there is slight variation in L 2 location.
Background: The location and stability of the equilibrium points are studied for the Planar Circu... more Background: The location and stability of the equilibrium points are studied for the Planar Circular Restricted Three-Body Problem where the more massive primary is an oblate spheroid. Methods: The mean motion of the equations of motion is formulated from the secular perturbations as derived by(1) and used in(2–4). The singularities of the equations of motion are found for locating the equilibrium points. Their stability is analysed using the linearized variational equations of motion at the equilibrium points. Findings: As the effect of oblateness in the mean motion expression increases, the location and stability of the equilibrium points are affected by the oblateness of the more massive primary. It is interesting to note that all the three collinear points move towards the more massive primary with oblateness. It is a new result. Among the shifts in the locations of the five equilibrium points, the y–location of the triangular equilibrium points relocate the most. It is very int...
Heat transfer in mixed convection unsteady MHD flow of an incompressible nanofluid in a channel u... more Heat transfer in mixed convection unsteady MHD flow of an incompressible nanofluid in a channel under the influence of heat source is studied. The channel with non-uniform walls temperature is taken in a perpendicular direction with a transverse magnetic field. Based on the substantial boundary conditions, three different flow conditions are examined. The problem is formed in PDEs with substantial boundary conditions. Four different forms of nanoparticles of identical volume fraction are employed in traditional base fluid water (H 2 O). Solutions for momentum and energy are attained by the perturbation method and examined graphically in different graphs. It is established that viscosity and thermal conductivity are the mainly well-known variables accountable for different results of velocity and temperature. It is also found that increasing heat source leads to an increase in nanofluid velocity and temperature and nano-size particles instance platelet and blade shapes have lesser mo...
The triangular Lagrangian points of the elliptic restricted three-body problem (ERTBP) with oblat... more The triangular Lagrangian points of the elliptic restricted three-body problem (ERTBP) with oblate and radiating more massive primary are studied. The mean motion equation used here is different from the ones employed in many studies on the perturbed ERTBP. The effect of oblateness on the mean motion equation varies. This change influences the location and stability of the triangular Lagrangian points. The points tend to shift in the y-direction. The influence of the oblateness on the critical mass ratio is also altered. But the eccentricity limit for stability remains the same.
International Journal of Astronomy and Astrophysics, 2011
An analytical theory for calculating perturbations of the orbital elements of a satellite due to ... more An analytical theory for calculating perturbations of the orbital elements of a satellite due to J 2 to accuracy up to fourth power in eccentricity is developed. It is observed that there is significant improvement in all the orbital elements with the present theory over second-order theory. The theory is used for computing the mean orbital elements, which are found to be more accurate than the existing Bhatnagar and Taqvi's theory (up to second power in eccentricity). Mean elements have a large number of practical applications.
The combination of atmospheric drag and lunar and solar perturbations in addition to Earth’s obla... more The combination of atmospheric drag and lunar and solar perturbations in addition to Earth’s oblateness influences the orbital lifetime of an upper stage in geostationary transfer orbit (GTO). These high eccentric orbits undergo fluctuations in both perturbations and velocity and are very sensitive to the initial conditions. The main objective of this paper is to predict the reentry time of the upper stage of the Indian geosynchronous satellite launch vehicle, GSLV-D5, which inserted the satellite GSAT-14 into a GTO on January 05, 2014, with mean perigee and apogee altitudes of 170 km and 35975 km. Four intervals of near linear variation of the mean apogee altitude observed were used in predicting the orbital lifetime. For these four intervals, optimal values of the initial osculating eccentricity and ballistic coefficient for matching the mean apogee altitudes were estimated with the response surface methodology using a genetic algorithm. It was found that the orbital lifetime from...
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1992
Analytical theory for the motion of near-Earth satellite orbits with the air drag effect is devel... more Analytical theory for the motion of near-Earth satellite orbits with the air drag effect is developed in terms of the KS elements, utilizing an analytical oblate exponential atmospheric model. The series expansions include up to cubic terms in e (eccentricity) and c (a small parameter dependent on the flattening of the atmosphere). Due to the symmetry of the KS element equations, only one of the eight equations is integrated analytically to obtain the state vector at the end of each revolution. Numerical comparisons are made with nine test cases, selected to cover a wide range of eccentricity with perigee heights near to 300 km at three different inclinations. A comparison of three orbital parameters: semi-major axis, eccentricity and argument of perigee, perturbed by air drag with oblate atmosphere is made with the previously developed second-order theory. It is found that with the present theory with increase in eccentricity there is improvement in semi-major axis and eccentricity...
Poincaré surface of section technique is used to study the evolution of a family 'f ' of simply s... more Poincaré surface of section technique is used to study the evolution of a family 'f ' of simply symmetric retrograde periodic orbits around the smaller primary in the framework of restricted three-body problem for a number of systems, actual and hypothetical, with mass ratio varying from 10 −7 to 0.015. It is found that as the mass ratio decreases the region of phase space containing the two separatrices shrinks in size and moves closer to the smaller primary. Also the corresponding value of Jacobi constant tends towards 3.
It is well known (cf. Szebehely, 1967a) that, in the restricted three-body problem, the condition... more It is well known (cf. Szebehely, 1967a) that, in the restricted three-body problem, the condition for linear stability of the two equidistant points of equilibrium is that the mass parameter # is less than the critical value
The motion of a particle in the restricted three-body problem is explored by treating the more ma... more The motion of a particle in the restricted three-body problem is explored by treating the more massive primary as an oblate spheroid with its equatorial plane coincident with the plane of motion of the primaries using a perturbation method. Initial conditions for the infinitesimal periodic orbits around the more massive primary are generated and the effect of oblateness on the perigee of these orbits is studied as well. It is observed that when oblateness coefficient is increased, the perigee of the orbit shifts towards both the primaries depending upon the increase in period and mass ratio. It is further noticed that during this transition, for certain periods, the perigee of the orbit remains unaltered with the increase in the oblateness coefficient.
A new non-singular analytical theory with respect to the Earth’s zonal harmonic terms J2, J3, J4 ... more A new non-singular analytical theory with respect to the Earth’s zonal harmonic terms J2, J3, J4 has been developed for short-periodic motion, by analytically integrating the uniformly regular KS canonical equations of motion using generalized eccentric anomaly ‘E’ as the independent variable. Only one of the eight equations need to be integrated analytically to generate the state vector, due to the symmetry in the equations of motion, and the computation for the other equations is done by changing the initial conditions. King-Hele’s expression for radial distance ‘r’ with J2 is also considered in generating the solution. The results obtained from the analytical expressions in a single step during half a revolution match quite well with numerically integrated values. Numerical results also indicate that the solution is reasonably accurate for a wide range of orbital elements during half a revolution and is an improvement over Sharon et al. [17] theory, which is generated in terms of...
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Papers by Ram Sharma