Training Aerospace Engineering Orbital Mechanics — Kepler & Newton
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Orbital Mechanics — Kepler & Newton

24 min Aerospace Engineering

Orbital Mechanics — Kepler & Newton

Spacecraft trajectories are governed by Newton's law of gravitation and Kepler's laws of planetary motion.

Newton's Law of Gravitation

$$F = \frac{G M m}{r^2}$$

where $G = 6.674 \times 10^{-11}$ N·m²/kg², $M$ = central body mass, $m$ = satellite mass, $r$ = distance from center.

Circular Orbit Velocity

$$v = \sqrt{\frac{GM}{r}} = \sqrt{\frac{\mu}{r}}$$

where $\mu = GM$ is the gravitational parameter. For Earth: $\mu = 3.986 \times 10^{14}$ m³/s².

Kepler's Third Law

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

where $a$ = semi-major axis and $T$ = orbital period.

Example 1

Find the orbital velocity and period for a satellite in a circular orbit at altitude $h = 400$ km above Earth ($R_E = 6{,}371$ km).

$r = R_E + h = 6{,}771$ km $= 6.771 \times 10^6$ m

$$v = \sqrt{\frac{3.986 \times 10^{14}}{6.771 \times 10^6}} = 7{,}672 \text{ m/s} \approx 7.67 \text{ km/s}$$

$$T = \frac{2\pi r}{v} = \frac{2\pi \times 6.771 \times 10^6}{7{,}672} = 5{,}542 \text{ s} \approx 92.4 \text{ min}$$

Example 2

Find the altitude of a geostationary orbit ($T = 24$ hours).

$T = 86{,}400$ s. From Kepler's third law:

$$a = \left(\frac{\mu T^2}{4\pi^2}\right)^{1/3} = \left(\frac{3.986 \times 10^{14} \times 86{,}400^2}{4\pi^2}\right)^{1/3} = 42{,}164 \text{ km}$$

Altitude: $h = 42{,}164 - 6{,}371 = 35{,}793$ km

Example 3

What escape velocity is needed from Earth's surface?

$$v_{\text{esc}} = \sqrt{\frac{2\mu}{R_E}} = \sqrt{\frac{2 \times 3.986 \times 10^{14}}{6.371 \times 10^6}} = 11{,}186 \text{ m/s} \approx 11.2 \text{ km/s}$$

Practice Problems

1. Find the orbital velocity at $h = 200$ km above Earth.
2. The ISS orbits at roughly 408 km altitude. Find its orbital period.
3. Find the orbital velocity around the Moon ($\mu_{\text{Moon}} = 4.905 \times 10^{12}$, $R_{\text{Moon}} = 1{,}737$ km) at 100 km altitude.
4. A satellite has $T = 12$ hours. Find its semi-major axis.
5. Find escape velocity from Mars ($\mu = 4.283 \times 10^{13}$, $R = 3{,}390$ km).
6. An elliptical orbit has perigee $r_p = 6{,}600$ km and apogee $r_a = 42{,}000$ km. Find the semi-major axis.
7. Find the period of the elliptical orbit in #6.
8. Vis-viva equation: $v = \sqrt{\mu(2/r - 1/a)}$. Find velocity at perigee in #6.
9. A Hohmann transfer from a 200 km orbit to a 35{,}786 km orbit requires two burns. Find $\Delta v_1$ at the lower orbit.
10. How does orbital velocity change with altitude?
11. If the orbital radius doubles, by what factor does the period change?
12. Find the gravitational acceleration at $h = 400$ km altitude.
Show Answer Key

1. $r = 6{,}571$ km; $v = \sqrt{3.986 \times 10^{14}/6.571 \times 10^6} = 7{,}788$ m/s

2. $r = 6{,}779$ km; $T = 2\pi\sqrt{(6.779\times10^6)^3/3.986\times10^{14}} = 5{,}554$ s $\approx 92.6$ min

3. $r = 1{,}837$ km; $v = \sqrt{4.905\times10^{12}/1.837\times10^6} = 1{,}634$ m/s

4. $T = 43{,}200$ s; $a = (\mu T^2/4\pi^2)^{1/3} = 26{,}560$ km

5. $v = \sqrt{2 \times 4.283\times10^{13}/3.39\times10^6} = 5{,}027$ m/s

6. $a = (r_p + r_a)/2 = (6{,}600 + 42{,}000)/2 = 24{,}300$ km

7. $T = 2\pi\sqrt{a^3/\mu} = 37{,}729$ s $\approx 10.5$ hours

8. $v_p = \sqrt{3.986\times10^{14}(2/6.6\times10^6 - 1/2.43\times10^7)} = 9{,}876$ m/s

9. $v_{\text{circ}} = 7{,}784$ m/s; $a_{\text{transfer}} = (6{,}571 + 42{,}157)/2 = 24{,}364$ km; $v_{\text{transfer}} = 10{,}152$ m/s at perigee; $\Delta v_1 \approx 2{,}368$ m/s

10. $v \propto 1/\sqrt{r}$ — velocity decreases with altitude

11. $T \propto r^{3/2}$; factor $= 2^{3/2} = 2\sqrt{2} \approx 2.83$

12. $g = \mu/r^2 = 3.986\times10^{14}/(6.771\times10^6)^2 = 8.69$ m/s²

Aerodynamic Forces — Lift & Drag