GATE Aerospace Engineering Space Dynamics Syllabus

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.

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

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

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 $\Delta t$.

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).

What is the eccentricity vector $\vec{e}$ and what does it point toward?

$$\vec{e} = \frac{1}{\mu}\left(\vec{v}\times\vec{h}\right) - \frac{\vec{r}}{r}$$ Its magnitude is the eccentricity and it points from the focus toward perigee.

How is the flight path angle $\gamma$ defined and computed?

$\gamma$ is the angle between the velocity vector and the local horizontal (perpendicular to $\vec{r}$): $$\tan\gamma = \frac{e\sin\theta}{1 + e\cos\theta}$$

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.

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

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

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).

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 $\varepsilon = -\mu/(2a) > 0$.

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 $T = 2\pi\sqrt{a^{3}/\mu}$.

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 $5430\ \mathrm{s}$.

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.

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 $a_{t} = (r_{1}+r_{2})/2$.

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).

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 $v_p = \sqrt{\frac{\mu}{a}\frac{1+e}{1-e}}$.

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 $\frac{v_{p}}{v_{a}} = \frac{r_{a}}{r_{p}} = \frac{1+e}{1-e}$.

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.

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).

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 $\theta$.

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

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

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

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

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.

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.

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).

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

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

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 $r$.

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

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

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.

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.