Earth Satellites - Projection of Satellite

To launch an artificial satellite into orbit around Earth, we need to give it a specific horizontal velocity at a particular height. A satellite requires a minimum two-stage rocket because the first stage lifts it to the desired height, and the second stage gives it horizontal velocity. The exact horizontal velocity needed for a satellite to revolve in a stable circular orbit around Earth is called critical velocity or orbital velocity. This velocity depends on the height of the satellite and Earth's mass, not on the satellite's mass. Understanding critical velocity is essential for successful satellite deployment and mission planning.

The exact horizontal velocity of projection that must be given to a satellite at a certain height so that it can revolve in a circular orbit around the Earth is called the critical velocity or orbital velocity (v_c).

v_c = \[\sqrt{\frac{GM}{R+h}}\]

Where:

• v_c = critical velocity (m/s)
• G = gravitational constant (6.67 × 10⁻¹¹ N·m²/kg²)
• M = mass of the Earth (kg)
• R = radius of the Earth (km)
• h = height of satellite above Earth's surface (km)

The critical velocity is derived by equating the centripetal force (needed for circular motion) with the gravitational force (pulling the satellite toward Earth).

For a satellite to move in a stable circular orbit, the gravitational force must provide exactly the right amount of centripetal force. If the velocity is too low, the satellite will fall back to Earth. If it's too high, it will either enter an elliptical orbit or escape Earth's gravitational influence entirely.

Various possible orbits depending on the value of v_h.

Case (I): v_h < v_c (Velocity Less Than Critical Velocity)

• The satellite follows an elliptical orbit with the point of projection as the apogee (farthest point from Earth).
• If the satellite enters Earth's atmosphere during this elliptical path, air resistance causes energy loss, and the satellite spirals down and crashes.

Case (II): v_h = v_c (Velocity Equal to Critical Velocity)

• The satellite moves in a stable, circular orbit around Earth.
• This is the ideal condition for maintaining a stable orbit.

Case (III): v_c < v_h < v_e (Velocity Between Critical and Escape Velocity)

• The satellite follows an elliptical orbit with the point of projection as the perigee (point closest to Earth).
• The satellite remains in orbit but travels farther away at the apogee.

Case (IV): v_h = v_e (Velocity Equal to Escape Velocity)

• The satellite travels along a parabolic path.
• It escapes Earth's gravitational influence but never returns to the initial point of projection.
• Its speed becomes zero at infinity.

Case (V): v_h > v_e (Velocity Greater Than Escape Velocity)

• The satellite follows a hyperbolic path.
• It completely escapes Earth's gravitational influence.

Satellite Close to Earth's Surface

When a satellite revolves very close to Earth's surface, the height (h) is negligible compared to Earth's radius (R). Therefore, we can assume R >> h, so (R + h) ≈ R.

v_c = \[\sqrt{\frac{GM}{R}}\]

Since g = \[\frac {GM}{R^2}\], we have GM = gR²

Therefore:

v_c = \[\sqrt{\frac{gR^{2}}{R}}=\sqrt{gR}\] = 7.92 km/s

This is the maximum possible critical speed and is approximately 25 times faster than the fastest passenger aeroplanes.

Important Facts:

• Critical velocity is independent of the satellite's mass—it depends only on Earth's mass and the orbital height.
• Critical velocity decreases with an increase in the height of the satellite.
• At Earth's surface, the critical velocity is 7.92 km/s.

• Satellite Launch Planning: Critical velocity determines the exact speed required to place a satellite in a stable orbit at a specific altitude.
• Mission Success: Achieving the correct critical velocity is essential for satellite stability; incorrect velocity leads to orbit failure or satellite loss.
• Fuel Efficiency: Understanding critical velocity helps rocket engineers calculate the minimum fuel needed for orbital insertion.
• Orbit Predictability: Critical velocity allows mission planners to predict satellite paths and orbital periods accurately.
• Independence from Satellite Mass: The formula shows that critical velocity depends only on orbital height and Earth's properties, making it applicable to all satellites regardless of their size or mass.
• Safety Consideration: Knowledge of critical velocity helps avoid escape velocities unintentionally, which would cause satellite loss.