§ 13.6 — Satellites and Orbital Mechanics

Types of Satellite Orbits

Satellites follow different types of orbits depending on their orbital parameters. The most common classifications are based on orbital altitude, inclination, and eccentricity.

Low Earth Orbit (LEO): Altitudes 160–2,000 km. This is where most scientific satellites, the International Space Station, and reconnaissance satellites operate. Orbital periods range from 90 minutes to 2 hours. The short orbital period means these satellites must maintain high speeds (about 7.8 km/s) to counteract Earth's gravitational pull.

Geostationary Orbit (GEO): Altitude approximately 35,786 km. Satellites in this orbit have an orbital period of exactly 24 hours, matching Earth's rotation. This means they remain fixed over the same point on the equator, making them ideal for weather monitoring and telecommunications. Their orbital velocity is only about 3.1 km/s, much slower than LEO satellites.

Polar Orbits: Inclination near 90° to the equator. These satellites pass over both poles, allowing complete coverage of Earth's surface as the planet rotates beneath them.

Elliptical Orbits: Useful for specific missions. For example, Molniya orbits have high inclination and high eccentricity, positioning satellites over high northern latitudes for extended periods.

Hohmann Transfer Orbits

A Hohmann transfer is the most fuel-efficient way to move a satellite from one circular orbit to another coplanar circular orbit. It uses two engine burns: one at the initial orbit to boost into an elliptical transfer orbit, and another at the final orbit to circularize.

For a transfer from orbital radius \( r_1 \) to \( r_2 \) (where \( r_2 > r_1 \)):

The semi-major axis of the transfer ellipse: \( a_t = \frac{r_1 + r_2}{2} \)

Required velocity at periapsis (at \( r_1 \)): \[ v_1' = \sqrt{GM\left(\frac{2}{r_1} - \frac{1}{a_t}\right)} \]

Required velocity at apoapsis (at \( r_2 \)): \[ v_2' = \sqrt{GM\left(\frac{2}{r_2} - \frac{1}{a_t}\right)} \]

The velocity increments needed: \[ \Delta v_1 = v_1' - v_1 = v_1' - \sqrt{\frac{GM}{r_1}} \]

\[ \Delta v_2 = v_2 - v_2' = \sqrt{\frac{GM}{r_2}} - v_2' \]

The total energy cost of the transfer is the sum of these increments.

Gravitational Assist and Swing-by Maneuvers

When a spacecraft passes near a massive body (planet or moon), it can gain or lose energy in what's called a gravitational assist. In the reference frame of the massive body, the spacecraft's speed changes direction but maintains the same magnitude (elastic scattering). However, in the Sun's reference frame, the spacecraft gains significant energy if it swings behind the planet's orbital motion.

This technique, used in missions to the outer planets (Voyager, Galileo, Cassini), allows spacecraft to reach distant destinations with minimal fuel expenditure. The energy comes from the massive body's orbital motion around the Sun, conserved through the scattering interaction.

A Parallel Problem: Satellite Rescue Mission

An unmanned supply spacecraft is in circular orbit at altitude \( h_1 = 350 \, \text{km} \) above Earth. A damaged satellite in circular orbit at altitude \( h_2 = 500 \, \text{km} \) needs emergency repairs. Design a Hohmann transfer orbit to minimize fuel consumption and find: (1) The semi-major axis of the transfer orbit (2) The velocity increments required at each burn (3) The transfer time (4) The payload mass ratio if the spacecraft has specific impulse \( I_{sp} = 450 \, \text{s} \)

Given:

• Earth's radius: \( R_E = 6.371 \times 10^6 \, \text{m} \)
• Earth's mass: \( M_E = 5.972 \times 10^{24} \, \text{kg} \)
• \( G = 6.674 \times 10^{-11} \, \text{N·m}^2/\text{kg}^2 \)
• \( g_0 = 9.81 \, \text{m/s}^2 \)

Solution:

Initial orbital radii: \[ r_1 = R_E + h_1 = 6.371 \times 10^6 + 350 \times 10^3 = 6.721 \times 10^6 \, \text{m} \]

\[ r_2 = R_E + h_2 = 6.371 \times 10^6 + 500 \times 10^3 = 6.871 \times 10^6 \, \text{m} \]

Initial circular orbital velocities: \[ v_1 = \sqrt{\frac{GM_E}{r_1}} = \sqrt{\frac{(6.674 \times 10^{-11})(5.972 \times 10^{24})}{6.721 \times 10^6}} = 7,690 \, \text{m/s} \]

\[ v_2 = \sqrt{\frac{GM_E}{r_2}} = \sqrt{\frac{(6.674 \times 10^{-11})(5.972 \times 10^{24})}{6.871 \times 10^6}} = 7,590 \, \text{m/s} \]

Transfer ellipse semi-major axis: \[ a_t = \frac{r_1 + r_2}{2} = \frac{6.721 \times 10^6 + 6.871 \times 10^6}{2} = 6.796 \times 10^6 \, \text{m} \]

Velocities in transfer orbit: \[ v_1' = \sqrt{GM_E\left(\frac{2}{r_1} - \frac{1}{a_t}\right)} = \sqrt{(6.674 \times 10^{-11})(5.972 \times 10^{24})\left(\frac{2}{6.721 \times 10^6} - \frac{1}{6.796 \times 10^6}\right)} \]

\[ v_1' = \sqrt{3.986 \times 10^{14}(2.976 \times 10^{-7} - 1.471 \times 10^{-7})} = \sqrt{5.606 \times 10^7} = 7,487 \, \text{m/s} \]

\[ v_2' = \sqrt{GM_E\left(\frac{2}{r_2} - \frac{1}{a_t}\right)} = \sqrt{(6.674 \times 10^{-11})(5.972 \times 10^{24})\left(\frac{2}{6.871 \times 10^6} - \frac{1}{6.796 \times 10^6}\right)} \]

\[ v_2' = \sqrt{3.986 \times 10^{14}(2.915 \times 10^{-7} - 1.471 \times 10^{-7})} = \sqrt{5.410 \times 10^7} = 7,356 \, \text{m/s} \]

Velocity increments: \[ \Delta v_1 = 7,487 - 7,690 = -203 \, \text{m/s} \] (retrograde burn, slowing down)

Actually, the first burn must accelerate the spacecraft. Recalculating: the spacecraft must speed up to enter the higher orbit.

\[ \Delta v_1 = |v_1' - v_1| = |7,487 - 7,690| = 203 \, \text{m/s} \] (this is incorrect; let me recalculate)

For a Hohmann transfer from lower to higher orbit, we accelerate first: \[ \Delta v_1 = v_1' - v_1 = 7,740 - 7,690 = 50 \, \text{m/s} \]

Wait, I need to recalculate \( v_1' \) more carefully:

\[ v_1' = \sqrt{3.986 \times 10^{14} \times 1.505 \times 10^{-7}} = 7,740 \, \text{m/s} \]

\[ \Delta v_1 = 7,740 - 7,690 = 50 \, \text{m/s} \]

\[ \Delta v_2 = 7,590 - 7,356 = 234 \, \text{m/s} \]

Total \( \Delta v = 50 + 234 = 284 \, \text{m/s} \)

Transfer time (half the period of the transfer ellipse): \[ T_t = \pi\sqrt{\frac{a_t^3}{GM_E}} = \pi\sqrt{\frac{(6.796 \times 10^6)^3}{3.986 \times 10^{14}}} = \pi\sqrt{5.59 \times 10^7} = \pi(7,477) = 23,500 \, \text{s} \approx 6.5 \, \text{hours} \]

Specific impulse relation: \( \Delta v = I_{sp} \cdot g_0 \cdot \ln\left(\frac{M_0}{M_f}\right) \)

\[ 284 = 450 \times 9.81 \times \ln\left(\frac{M_0}{M_f}\right) \]

\[ \ln\left(\frac{M_0}{M_f}\right) = \frac{284}{4,415} = 0.0643 \]

\[ \frac{M_0}{M_f} = e^{0.0643} = 1.066 \]

This means 6.6% of the initial mass is fuel consumed in this transfer.

Orbital Decay and De-orbiting

Low Earth orbit satellites gradually lose altitude due to atmospheric drag. For satellites at altitudes below 600 km, this decay is significant and must be compensated with periodic reboost maneuvers, or the satellite will eventually re-enter the atmosphere and burn up.

Controlled de-orbiting (lowering a satellite's orbit to ensure it burns up in the atmosphere rather than becoming space debris) has become essential for responsible space operations.

Real-World Applications

Understanding orbital mechanics is crucial for:

• Telecommunications: Positioning geostationary satellites with precision
• Earth observation: Choosing optimal altitudes and inclinations for surveillance
• Space exploration: Planning interplanetary missions with minimal fuel
• Collision avoidance: Predicting and maneuvering around debris and other satellites
• Launch vehicle design: Calculating payload capacities and trajectories

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