GATE Aerospace Engineering Space Dynamics Syllabus

Space Dynamics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Space Dynamics in GATE Aerospace Engineering, not a summary of it.

1. 1

   Central force motion

   2 topics

   • Determination of Trajectory
   • Orbital Period in Simple Cases
2. 2

   Kepler’s Laws

   overview

   Examined as a single unit within Space Dynamics — no further topic split in the official outline.

3. 3

   Escape Velocity

   overview

   Examined as a single unit within Space Dynamics — no further topic split in the official outline.

Space Dynamics flashcards for GATE Aerospace Engineering

24 of 50 cards from the Space Dynamics deck — real questions with worked answers.

1. What is the standard gravitational parameter $\mu$ for a two-body system, and how is it defined?

   $\mu$ is the product of the universal gravitational constant and the central body's mass: $\mu = G M$. For Earth, $\mu \approx 3.986 \times 10^{14}\ \mathrm{m^3/s^2}$.

2. State Newton's law of universal gravitation in vector form for the relative motion of two bodies.

   $$\ddot{\vec{r}} = -\frac{\mu}{r^{3}}\vec{r}$$ where $\vec{r}$ is the position of the orbiting body relative to the central body and $r = |\vec{r}|$.

3. What is the general equation of a conic-section orbit (the trajectory equation) in polar form?

   $$r = \frac{p}{1 + e\cos\theta}$$ where $p$ is the semi-latus rectum, $e$ is the eccentricity, and $\theta$ is the true anomaly.

4. How is the semi-latus rectum $p$ related to the specific angular momentum $h$?

   $$p = \frac{h^{2}}{\mu}$$

5. Classify the four conic-section orbits by their eccentricity $e$.

   Circle: $e = 0$; Ellipse: $0 < e < 1$; Parabola: $e = 1$; Hyperbola: $e > 1$.

6. What is the vis-viva equation relating orbital speed to position?

   $$v^{2} = \mu\left(\frac{2}{r} - \frac{1}{a}\right)$$ where $a$ is the semi-major axis.

7. What is the specific orbital energy $\varepsilon$ and how does it relate to the semi-major axis?

   $$\varepsilon = \frac{v^{2}}{2} - \frac{\mu}{r} = -\frac{\mu}{2a}$$ It is constant along the orbit.

8. What is the sign of specific orbital energy for elliptical, parabolic, and hyperbolic trajectories?

   Elliptical: $\varepsilon < 0$; Parabolic: $\varepsilon = 0$; Hyperbolic: $\varepsilon > 0$.

9. State Kepler's third law for the orbital period of an elliptical orbit.

   $$T = 2\pi\sqrt{\frac{a^{3}}{\mu}}$$ The period depends only on the semi-major axis $a$.

10. For a circular orbit of radius $r$, what is the orbital speed?

    $$v_{c} = \sqrt{\frac{\mu}{r}}$$

11. For a circular orbit of radius $r$, what is the orbital period?

    $$T = 2\pi\sqrt{\frac{r^{3}}{\mu}}$$

12. What is the specific angular momentum $\vec{h}$ in a two-body orbit, and why is it conserved?

    $$\vec{h} = \vec{r}\times\vec{v}$$ It is conserved because gravity is a central force (torque about the focus is zero), so the orbit lies in a fixed plane.

13. State Kepler's second law (law of areas).

    The radius vector sweeps out equal areas in equal times; the areal velocity $\frac{dA}{dt} = \frac{h}{2}$ is constant.

14. How do you express the period of an ellipse using the area-swept rate and total area?

    $$T = \frac{\text{Area}}{dA/dt} = \frac{\pi a b}{h/2} = \frac{2\pi a b}{h}$$ which reduces to $2\pi\sqrt{a^{3}/\mu}$.

15. What is the relationship between the semi-minor axis $b$, semi-major axis $a$, and eccentricity $e$ of an ellipse?

    $$b = a\sqrt{1 - e^{2}}$$

16. Give the radii of perigee and apogee in terms of $a$ and $e$.

    Perigee: $r_{p} = a(1 - e)$; Apogee: $r_{a} = a(1 + e)$.

17. How is the semi-major axis recovered from perigee and apogee radii?

    $$a = \frac{r_{p} + r_{a}}{2}$$

18. How is eccentricity expressed in terms of apogee and perigee radii?

    $$e = \frac{r_{a} - r_{p}}{r_{a} + r_{p}}$$

19. What is Kepler's equation relating mean anomaly $M$ and eccentric anomaly $E$?

    $$M = E - e\sin E$$

20. Define the mean anomaly $M$ in terms of mean motion $n$ and time since perigee.

    $$M = n(t - t_{p}) = nt - M_{0}$$ where the mean motion $n = \sqrt{\mu/a^{3}} = 2\pi/T$.

21. What is the mean motion $n$ and its relation to the orbital period?

    $$n = \sqrt{\frac{\mu}{a^{3}}} = \frac{2\pi}{T}$$ It is the average angular rate over one orbit.

22. State the relation between true anomaly $\theta$ and eccentric anomaly $E$.

    $$\tan\frac{\theta}{2} = \sqrt{\frac{1+e}{1-e}}\,\tan\frac{E}{2}$$

23. What are the steps to determine the time of flight between two points on an elliptic orbit?

    1) Find true anomaly $\theta$ at each point; 2) Convert to eccentric anomaly $E$; 3) Use Kepler's equation $M = E - e\sin E$ to get mean anomaly; 4) Compute $t = M/n$ at each point; 5) Take the difference $\Delta t$.

24. In determining a trajectory, what two state vectors fully specify an orbit at an instant?

    The position vector $\vec{r}$ and the velocity vector $\vec{v}$ at one instant completely determine the orbit (six elements from six components).

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