JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech III Year SPACE PROPULSION Previous Question Paper (August - 2022 - R18)

Orbits are classified based on their shape, altitude, and inclination. The fundamental orbital types include:

• Low Earth Orbit (LEO): Typically between 160 km and 2,000 km altitude. Used for Earth observation and the International Space Station.
• Medium Earth Orbit (MEO): Located between LEO and Geostationary orbit, commonly used for navigation satellite constellations like GPS.
• Geostationary Orbit (GEO): A circular orbit at approximately 35,786 km altitude above the equator, where the orbital period matches Earth's rotation, keeping the satellite fixed over a single point.
• Highly Elliptical Orbit (HEO): Characterized by a high eccentricity, such as the Molniya orbit, which provides long dwell times over high-latitude regions.
• Sun-Synchronous Orbit (SSO): A near-polar orbit that maintains a constant orientation relative to the Sun, useful for consistent lighting conditions in imaging.

The motion of a satellite is governed by the two-body problem, derived from Newton's Law of Universal Gravitation. The trajectory is described by the polar equation for a conic section:

Where:

• r is the radial distance from the central body.
• p is the semi-latus rectum, defined as p = h² / μ, where h is the specific angular momentum and μ is the gravitational parameter.
• e is the eccentricity, which determines the orbit shape: e=0 (circular), 0<e<1 (elliptical), e=1 (parabolic), and e>1 (hyperbolic).
• θ is the true anomaly, the angle between the satellite's position and the periapsis.

The orbital period T for an elliptical orbit is given by Kepler's Third Law:

Where a is the semi-major axis. These equations define the geometry and timing of a satellite's path through space.

The Tsiolkovsky rocket equation, also known as the ideal rocket equation, is a fundamental mathematical relationship that describes the motion of vehicles that follow the basic principle of a rocket: a device that can apply acceleration to itself using thrust by expelling part of its mass with high velocity. The equation is derived from the conservation of linear momentum in an isolated system.

The equation is expressed as:

Where:

• Δv is the maximum change of velocity of the vehicle (delta-v).
• v_e is the effective exhaust velocity, calculated as the product of the specific impulse (I_sp) and the acceleration due to gravity (g₀).
• m₀ is the initial total mass of the vehicle, including propellant.
• m_f is the final total mass of the vehicle after propellant is consumed.
• ln is the natural logarithm function.

The term m₀ / m_f is known as the mass ratio. This equation demonstrates that the change in velocity is directly proportional to the exhaust velocity and the natural logarithm of the mass ratio, highlighting the exponential nature of rocket performance requirements.

Regarding its applicability, the equation provides an ideal theoretical limit for rocket performance. It assumes the following conditions:

1. The rocket is operating in a vacuum, meaning there is no atmospheric drag.
2. There are no external forces acting on the rocket, such as gravity or aerodynamic forces.
3. The exhaust velocity is constant throughout the burn.

In practical applications, the equation must be modified to account for losses. When operating within a planetary atmosphere, the effective delta-v is reduced by gravity losses (due to the time taken to gain altitude) and aerodynamic drag losses. Consequently, engineers use the equation as a baseline for mission planning, such as calculating the propellant mass fraction required to reach a specific orbit, while adding margins to compensate for these real-world environmental factors.