Orbital Energy and Equilibrium in Space (Main Topic 5)

Bound Orbit

• When the total energy (E) is negative

• Meaning the satellite is gravitationally bound to a planet or mass

• Closed, either circular or elliptical

Parabolic & Hyperbolic

These trajectories have positive energy, meaning the object escapes the gravitational field.

Lagrange Points

In the restricted three body problem, a small object of negligible mass can remain in a fixed position relative to two more massive orbiting bodies if placed at specific locations where the gravitational and centrifugal forces balance.

Lagrange Point 1 (L₁)

• This lies along the line connecting the two massive bodies, between them.

• This is commonly used for solar observatories, such as the SOHO spacecraft, because it provides an uninterrupted view of the Sun.

Lagrange Point 2 (L₂)

Ideal for deep space telescopes, such as the James Webb Space Telescope, because it offers a stable thermal environment and minimal interference from the Sun or Earth.

Lagrange Point 3 (L₃)

This lies on the opposite side of the larger body, directly opposite the smaller one. It is the least accessible of the colinear points and has no current practical use in mission design.

Lagrange Point 4 & 5 (L_{4 & 5})

• These points form equilateral triangles with the two massive bodies and lie ahead of and behind the smaller body in its orbit, respectively

• These points are stable for systems in which the mass ratio between the primary bodies exceeds approximately 24.96, as in the Sun and Sun Jupiter systems

Halo Orbit

• It is a three-dimensional periodic orbit around a collinear Lagrange point such as L₁ or L₂ Lagrange Point.

• Instead of staying exactly at the unstable point, a spacecraft orbits around it in a large looping path above and below the orbital plane.

Lissajous orbit

• It is a quasi-periodic orbit around a collinear Lagrange point.

• The spacecraft follows a complex looping pattern caused by oscillations in multiple directions.

• Unlike halo orbits, the path does not exactly repeat, creating a more flexible trajectory.

• Requires less precise insertion compared to halo orbits but still needs small station keeping maneuvers.

Lagrange Points

They are significant in mission planning, as they enable spacecraft to remain in stable or quasi stable positions relative to the Earth and Sun with minimal propellant usage, thereby supporting continuous observation, communication, and deep space exploration objectives