Orbital Mechanics

Orbital mechanics, also known as astrodynamics, is the application of ballistics and celestial mechanics to the motion of rockets and other spacecraft. Understanding orbital mechanics is crucial for space missions, satellite operations, and interplanetary travel.

Historical Foundation: Kepler's Laws

Johannes Kepler formulated three fundamental laws that describe planetary motion. These laws also apply to artificial satellites and spacecraft.

Kepler's First Law: Law of Ellipses

The orbit of every planet is an ellipse with the Sun at one of the two foci.

Mathematically, the equation of an elliptical orbit in polar coordinates is:

Where:

• $r$ = distance from the central body
• $a$ = semi-major axis
• $e$ = eccentricity (0 for circle, 1 for parabola)
• $\nu$ = true anomaly (angle from periapsis)

Kepler's Second Law: Law of Equal Areas

A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.

This law means that objects in elliptical orbits move faster when they are closer to the central body (periapsis) and slower when farther away (apoapsis).

Kepler's Third Law: Law of Harmonies

The square of the orbital period is directly proportional to the cube of the semi-major axis:

Where:

• $T$ = orbital period
• $a$ = semi-major axis
• $\mu$ = gravitational parameter ($GM$)

Orbital Elements

Six parameters, called orbital elements, completely define an orbit:

1. Semi-major axis (a): Defines the size of the orbit
2. Eccentricity (e): Defines the shape (0 for circular, 0-1 for elliptical)
3. Inclination (i): Tilt of the orbital plane relative to a reference plane
4. Right ascension of the ascending node (Ω): Orientation of the orbital plane
5. Argument of periapsis (ω): Orientation of the ellipse within the orbital plane
6. True anomaly (ν): Position of the object in its orbit at a specific time

The Two-Body Problem

The motion of two bodies interacting gravitationally can be solved analytically:

Where $\vec{r}$ is the position vector from the central body to the orbiting body.

Orbital Velocity

The velocity of an object in an elliptical orbit is given by the vis-viva equation:

Where:

• $v$ = orbital velocity
• $\mu$ = gravitational parameter
• $r$ = radial distance
• $a$ = semi-major axis

Types of Orbits

Low Earth Orbit (LEO)

• Altitude: 160-2,000 km
• Period: ~90 minutes
• Applications: ISS, Earth observation, Hubble Space Telescope

Geostationary Orbit (GEO)

• Altitude: 35,786 km
• Period: 24 hours (synchronous with Earth's rotation)
• Applications: Communications, weather satellites

Polar Orbit

• Inclination: ~90°
• Passes over Earth's poles
• Applications: Earth observation, reconnaissance

Orbital Maneuvers

Hohmann Transfer

The most fuel-efficient way to transfer between two circular orbits is the Hohmann transfer, which uses an elliptical transfer orbit:

Where $r_1$ and $r_2$ are the radii of the initial and final orbits.

Orbital Inclination Change

Changing orbital plane is expensive in terms of delta-v:

Where $\Delta i$ is the change in inclination and $v$ is the orbital velocity.

Real-World Application: GPS Satellite Constellation

The Global Positioning System uses a constellation of satellites in Medium Earth Orbit (MEO) at approximately 20,200 km altitude.

GPS Orbit Parameters

• Semi-major axis: ~26,560 km
• Period: ~12 hours
• Inclination: 55°
• Number of satellites: 31 (as of 2023)

Example Calculation

Calculate the orbital velocity of a GPS satellite:

import math

# Constants
G = 6.67430e-11  # Gravitational constant (m³/kg·s²)
M_earth = 5.972e24  # Earth mass (kg)
R_earth = 6.371e6   # Earth radius (m)

# GPS satellite parameters
altitude = 20200e3  # 20,200 km in meters
r = R_earth + altitude  # orbital radius
a = r  # for circular orbit, semi-major axis equals radius

# Gravitational parameter
mu = G * M_earth

# Calculate orbital velocity using vis-viva equation (for circular orbit)
v_orbital = math.sqrt(mu / r)

print(f"GPS satellite orbital velocity: {v_orbital:.2f} m/s")
print(f"GPS satellite orbital velocity: {v_orbital/1000:.2f} km/s")
print(f"GPS satellite orbital period: {2*math.pi*r/v_orbital/3600:.2f} hours")

Your Challenge: Interplanetary Mission Design

Design a basic Hohmann transfer from Earth to Mars for a potential crewed mission.

Goal: Calculate the delta-v requirements and transfer time for an Earth-Mars mission.

Key Variables

# Planetary data (approximate circular orbits)
AU = 1.496e11  # Astronomical unit (m)
r_earth = 1.0 * AU  # Earth's orbital radius
r_mars = 1.524 * AU  # Mars' orbital radius

# Gravitational parameter of Sun
mu_sun = 1.327e20  # m³/s²

# Calculate transfer orbit parameters
a_transfer = (r_earth + r_mars) / 2  # Semi-major axis of transfer orbit

Calculate the delta-v needed for:

1. Earth departure burn (from Earth orbit to transfer orbit)
2. Mars arrival burn (from transfer orbit to Mars orbit)
3. Total mission delta-v

Also calculate the transfer time using Kepler's third law.

Hint:

• Use the vis-viva equation to find velocities in each orbit
• The transfer time is half the period of the elliptical transfer orbit

# TODO: Calculate delta-v and transfer time
delta_v_departure = 0  # Delta-v for Earth departure
delta_v_arrival = 0    # Delta-v for Mars arrival  
transfer_time = 0      # Total transfer time in days

# Print results
print(f"Earth departure delta-v: {delta_v_departure:.2f} m/s")
print(f"Mars arrival delta-v: {delta_v_arrival:.2f} m/s")
print(f"Total mission delta-v: {delta_v_departure + delta_v_arrival:.2f} m/s")
print(f"Transfer time: {transfer_time:.2f} days")

How would you improve this basic model to account for more realistic mission constraints?