GATE Aerospace Engineering Space Dynamics Flashcards

24 sample cards from the Space Dynamics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. At perigee and apogee, what is the value of the flight path angle $\gamma$?

   $\gamma = 0$ at both perigee and apogee, since velocity is purely tangential (perpendicular to the radius vector) at the apsides.

2. What is the escape velocity from a distance $r$ (parabolic speed)?

   $$v_{esc} = \sqrt{\frac{2\mu}{r}} = \sqrt{2}\,v_{c}$$

3. What is the hyperbolic excess velocity $v_{\infty}$ and how does it relate to energy?

   $v_{\infty}$ is the residual speed at infinity on a hyperbolic trajectory: $$v_{\infty}^{2} = v^{2} - v_{esc}^{2} = -\frac{\mu}{a} = 2\varepsilon$$ (with $a < 0$ for a hyperbola).

4. For a hyperbolic orbit, what is the sign of the semi-major axis $a$, and what is the energy?

   $a < 0$ for a hyperbola, and the specific energy $\varepsilon = -\mu/(2a) > 0$.

5. Compare the orbital period of a circular orbit and an elliptical orbit having the same semi-major axis.

   They are identical. Period depends only on $a$, so any orbit with the same semi-major axis ($r = a$ for circle) shares the same period $T = 2\pi\sqrt{a^{3}/\mu}$.

6. What is the period of a Low Earth Orbit at roughly $r \approx 6678\ \mathrm{km}$ (about 300 km altitude)?

   Approximately $90$ minutes. Using $T = 2\pi\sqrt{r^{3}/\mu}$ with $\mu_{Earth} = 3.986\times10^{14}\ \mathrm{m^3/s^2}$ gives roughly $5430\ \mathrm{s}$.

7. What is a geostationary orbit's period and approximate radius?

   Period equals one sidereal day, $T \approx 86164\ \mathrm{s}$ (about 23 h 56 min); radius $r \approx 42164\ \mathrm{km}$ from Earth's center.

8. What is the period of the Hohmann transfer ellipse, and what fraction is the transfer time?

   The transfer takes half a period of the transfer ellipse: $$t_{transfer} = \frac{T}{2} = \pi\sqrt{\frac{a_{t}^{3}}{\mu}}$$ where $a_{t} = (r_{1}+r_{2})/2$.

9. In a parabolic trajectory, what equation replaces Kepler's equation for time of flight (Barker's equation)?

   $$M_{p} = D + \frac{D^{3}}{3}$$ where $D = \tan(\theta/2)$ is the parabolic anomaly and $M_{p} = \sqrt{\mu/p^{3}}\,(t - t_{p})\cdot$ (with appropriate scaling); it is a cubic in $D$ (Barker's equation).

10. What is the speed at perigee of an ellipse in terms of $h$ and $r_{p}$?

    $$v_{p} = \frac{h}{r_{p}}$$ since velocity is purely tangential at perigee, and equivalently $v_p = \sqrt{\frac{\mu}{a}\frac{1+e}{1-e}}$.

11. How does the orbital speed vary between perigee and apogee?

    Speed is maximum at perigee and minimum at apogee. By conservation of angular momentum, $v_{p}r_{p} = v_{a}r_{a}$, so $\frac{v_{p}}{v_{a}} = \frac{r_{a}}{r_{p}} = \frac{1+e}{1-e}$.

12. What does the conservation of mechanical energy imply about how speed changes with radius in an orbit?

    From $\varepsilon = v^{2}/2 - \mu/r = \text{const}$, speed increases as $r$ decreases and decreases as $r$ increases; the body moves fastest closest to the focus.

13. What are the six classical (Keplerian) orbital elements?

    Semi-major axis $a$, eccentricity $e$, inclination $i$, right ascension of ascending node $\Omega$, argument of perigee $\omega$, and true anomaly $\theta$ (or mean anomaly).

14. Which orbital elements define the orbit's shape and size, versus its orientation?

    Shape and size: $a$ and $e$. Orientation in space: inclination $i$, RAAN $\Omega$, and argument of perigee $\omega$. Position along the orbit: true anomaly $\theta$.

15. How is true anomaly $\theta$ at a given radius $r$ found from the orbit equation?

    $$\cos\theta = \frac{1}{e}\left(\frac{p}{r} - 1\right) = \frac{1}{e}\left(\frac{a(1-e^{2})}{r} - 1\right)$$

16. What is the relation between specific angular momentum $h$, semi-major axis $a$, and eccentricity $e$?

    $$h = \sqrt{\mu a (1 - e^{2})} = \sqrt{\mu p}$$

17. Why does the orbital period not depend on the eccentricity of the orbit?

    Because Kepler's third law gives $T = 2\pi\sqrt{a^{3}/\mu}$, which contains only the semi-major axis; two orbits with the same $a$ but different $e$ have equal periods.

18. What is the period of a satellite in a circular orbit at the surface of a body of radius $R$ (minimum-radius orbit)?

    $$T = 2\pi\sqrt{\frac{R^{3}}{\mu}} = 2\pi\sqrt{\frac{R}{g}}$$ since $g = \mu/R^{2}$ at the surface.

19. For a near-surface circular orbit, express the period using surface gravity $g$ and radius $R$.

    $$T = 2\pi\sqrt{\frac{R}{g}}$$ For Earth ($R \approx 6371\ \mathrm{km}$, $g \approx 9.81\ \mathrm{m/s^2}$) this gives about 84.5 minutes (Schuler period).

20. What is the relationship between the eccentric anomaly $E$ and radius $r$ on an elliptic orbit?

    $$r = a(1 - e\cos E)$$

21. Given $\vec{r}$ and $\vec{v}$, how is the semi-major axis $a$ determined for trajectory determination?

    From the energy equation: $$a = \left(\frac{2}{r} - \frac{v^{2}}{\mu}\right)^{-1}$$ computed using the current speed $v$ and radius $r$.

22. How does the period scale if the semi-major axis is doubled?

    Since $T \propto a^{3/2}$, doubling $a$ multiplies the period by $2^{3/2} = 2\sqrt{2} \approx 2.83$.

23. What is the average orbital angular velocity (mean motion) of a geostationary satellite in degrees per hour?

    $$n = \frac{360^{\circ}}{T} = \frac{360^{\circ}}{23.934\ \mathrm{h}} \approx 15.04^{\circ}/\mathrm{h}$$ matching Earth's rotation rate.

24. Summarize the procedure to determine the full trajectory from initial position and velocity vectors.

    1) Compute $\vec{h} = \vec{r}\times\vec{v}$ and $r,v$; 2) Find energy $\varepsilon$ and hence $a = -\mu/(2\varepsilon)$; 3) Compute eccentricity vector $\vec{e}$ to get $e$; 4) Determine inclination from $\vec{h}$; 5) Find node vector for $\Omega$ and $\omega$; 6) Find true anomaly $\theta$ from $\vec{e}\cdot\vec{r}$. These six elements fully specify the orbit.

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