List of top Space Dynamics Questions on Orbital Mechanics asked in GATE AE

An earth satellite moves in an elliptical orbit with a perigee altitude of 300 km and an apogee altitude of 3000 km. Assume that the radius of the earth is 6378 km. The eccentricity of the orbit is ________ (rounded off to three decimal places).

An earth satellite has the instantaneous position vector $\vec{r}$ and velocity vector $\vec{v}$ as given below. Here $\hat{p}$ and $\hat{q}$ denote the unit vectors along the $x$ and $y$ axes of the perifocal frame, respectively. Assume that the value of the gravitational parameter is 398600 $\text{km}^3/\text{s}^2$. Which one of the following trajectories does the satellite follow?
\[ \vec{r} = (8000\hat{p} + 9000\hat{q}) \text{ km and } \vec{v} = (-6\hat{p} + 6\hat{q}) \text{ km/s} \]

F and G denote two points on a spacecraft’s orbit around a planet, as indicated in the figure. O is the center of the planet, P is the periapsis, and the angles are as indicated in the figure. If $ OF = 8000 \, {km} $, $ OG = 10000 \, {km} $, $ \theta_F = 0^\circ $, and $ \theta_G = 60^\circ $, the eccentricity of the spacecraft's orbit is __________ (rounded off to two decimal places).

F and G denote two points on a spacecraft’s orbit around a planet, as indicated in the figure. O is the center of the planet, P is the periapsis, and the angles are as indicated in the figure. If $ OF = 8000 \, {km} $, $ OG = 10000 \, {km} $, $ \theta_F = 0^\circ $, and $ \theta_G = 60^\circ $, the eccentricity of the spacecraft's orbit is ___________ (rounded off to two decimal places).

F and G denote two points on a spacecraft’s orbit around a planet, as indicated in the figure. O is the center of the planet, P is the periapsis, and the angles are as indicated in the figure. If $ OF = 8000 \, {km} $, $ OG = 10000 \, {km} $, $ \theta_F = 0^\circ $, and $ \theta_G = 60^\circ $, the eccentricity of the spacecraft's orbit is \_\_\_\_\_\_\_\_ (rounded off to two decimal places).
\begin{figure} \centering \includegraphics[width=0.5\linewidth]{53.png} \end{figure}

Which of the following statements is/are correct about a satellite moving in a geostationary orbit?

An unpowered space vehicle at altitude $600\ \text{km}$ (above mean sea level) is moving at speed $9\ \text{km s}^{-1}$ relative to an Earth-centered frame. Take Earth's radius $R_E=6400\ \text{km}$ and $GM_E=3.98\times10^{14}\ \text{m}^3\text{s}^{-2}$. The trajectory is:

Consider an artificial satellite moving around the Moon in an elliptic orbit. The altitude of the satellite from the Moon's surface at the perigee is 25 km and at the apogee is 134 km. Assume the Moon to be spherical with a radius of 1737 km. The trajectory is considered with reference to a coordinate system fixed to the center of mass of the Moon. The ratio of the speed of the satellite at the perigee to that at the apogee is .......... (rounded off to 2 decimal places).

Given $G=6.67\times10^{-11}\ \text{N\,m}^2\text{/kg}^2$, planet mass $M=6.4169\times10^{23}$ kg and radius $R=3390$ km, find the escape velocity (km/s). (round off to one decimal place)

A satellite is in a circular orbit around Earth with period $T=90$ minutes. Take Earth's radius $R_E=6370$ km, Earth's mass $M_E=5.98\times10^{24}$ kg, and $G=6.67\times10^{-11}\ \text{N\,m}^2\text{/kg}^2$. Find the altitude above mean sea level (km).

Which one of the following figures represents the qualitative variation of absolute deceleration $\left|\dfrac{dV}{dt}\right|$ with altitude $h$ (measured from mean sea level) for a space vehicle undergoing a ballistic entry into the Earth's atmosphere?

Consider a high Earth-orbiting satellite of angular momentum per unit mass $\vec{h}$ and eccentricity $e$. The mass of the Earth is $M$ and $G$ is the universal gravitational constant. The distance between the satellite’s center of mass and the Earth’s center of mass is $r$, the true anomaly is $\theta$, and the phase angle is zero. Which of the following statements is/are true?

Compute the velocity required from Earth's surface to reach a circular orbit at 250 km altitude (round off to two decimals).
Earth data: $ GM_e = 398600.4 \, \text{km}^3/\text{s}^2 $, $ R_0 = 6378.14 \, \text{km} $