4.21 Orbital Velocity

Introduction

Orbital velocity is the precise speed that an object, such as a satellite or a planet, must maintain to stay in a stable orbit around a larger celestial body. This velocity represents a perfect balance: the gravitational pull of the central body provides the exact amount of centripetal force required to keep the orbiting object continuously turning in its circular path. If an object moves too slowly, it will fall back to the central body; if it moves too fast, it will fly off into a higher orbit or escape the gravitational pull altogether.

1. The Balance of Forces in an Orbit

For an object of mass $m$ (e.g., a satellite) to maintain a stable circular orbit around a much larger body of mass $M$ (e.g., a planet), the two forces acting on it must be balanced:

• Gravitational Force ($F_g$): This is the attractive force pulling the satellite towards the center of the planet. According to Newton's Law of Universal Gravitation, it is given by: $F_g=\frac{GMm}{r^2}$

• Centripetal Force ($F_c$): This is the net force required to keep the satellite moving in a circle. It is not a separate force but the resultant force directed towards the center of the orbit. Its magnitude is: $F_c=\frac{m{v}^2}{r}$

For a stable orbit, the gravitational force is the centripetal force.

2. Derivation of the Orbital Velocity Formula

By setting the gravitational force equal to the required centripetal force, we can derive the formula for orbital velocity ($v$).

$F_g=F_c$ $\frac{GMm}{r^2}=\frac{m{v}^2}{r}$

We can simplify this equation:

1. Cancel the satellite's mass ($m$) from both sides.
2. Cancel one factor of the radius ($r$) from both sides.

This leaves us with: $\frac{GM}{r}=v^2$

Solving for the orbital velocity $v$: $\boxed{v=\sqrt{\frac{GM}{r}}}$

3. Key Insights from the Formula

The formula for orbital velocity reveals two crucial points:

• Independence from Satellite's Mass: The mass of the orbiting object ($m$) does not appear in the final equation. This means that a small satellite and a large space station will have the exact same orbital velocity if they are at the same orbital radius.

• Dependence on Central Body and Radius: The required orbital speed depends only on:

  • The mass of the central body ($M$).
  • The radius of the orbit ($r$), which is the distance from the center of the central body.

• Inverse Relationship with Radius: As the orbital radius ($r$) increases, the required orbital velocity ($v$) decreases. Satellites in higher orbits move more slowly than satellites in lower orbits.

For a satellite orbiting very close to the Earth's surface, where $r\approx R$ (radius of Earth) and $g=\frac{GM}{R^2}$, the formula simplifies to: $v_o=\sqrt{gR}\approx 7.9\text{ km/s}$

4. Weightlessness in Orbiting Satellites

Astronauts inside an orbiting satellite experience apparent weightlessness. This is not because gravity is absent — gravity is very much present and is, in fact, the force keeping the satellite in orbit.

The reason for weightlessness is that both the satellite and the astronauts inside it are in continuous free fall toward Earth at exactly the same rate. Since there is no relative acceleration between the astronaut and the satellite floor, the floor exerts no normal (contact) force on the astronaut. It is this normal force that we perceive as weight, so when it vanishes, the astronaut feels weightless.

This is analogous to a person in a freely falling lift: as the lift accelerates downward at $g$, the person inside feels weightless because the floor no longer pushes up on them.

Summary Table

Possible Questions and Answers

• Q: What would happen if a satellite in a stable circular orbit suddenly slowed down?

• A: If its speed dropped below the required orbital velocity for that altitude, the gravitational force would become greater than the centripetal force needed to maintain the circle. The satellite would begin to lose altitude and spiral back towards Earth.

• Q: Why do satellites in higher orbits move slower?

• A: At a greater distance (larger $r$), the force of gravity is weaker. According to the formula $v=\sqrt{GM/r}$, since the gravitational pull is less, a smaller speed is required to maintain the balance needed for a stable orbit.

• Q: Why do astronauts feel weightless in an orbiting satellite?

• A: Both the satellite and the astronauts are in free fall toward Earth with the same acceleration. There is no contact (normal) force between the astronaut and the satellite floor, so the astronaut experiences apparent weightlessness.

Dimensions→

This principle is fundamental to celestial mechanics and space exploration, dictating the conditions required to place and maintain satellites, space stations, and probes in orbit around Earth and other celestial bodies.