7.7P Explain how, for a stable orbit, the radius must change if orbital…

The idea

A body stays in a stable orbit when gravity supplies exactly the centripetal force it needs. For a satellite of mass m moving at speed v on a circle of radius r around a body of mass M, the required force is mv²/r and gravity provides GMm/r². Setting these equal gives v² = GM/r, so speed and radius are linked: a larger radius means a smaller orbital speed, and a smaller radius means a larger orbital speed. The reasoning is qualitative here: if a satellite's orbital speed increases, it can no longer stay on the same circle, so to remain stable its orbit must move to a smaller radius; if its speed decreases, the stable orbit must be at a larger radius. The mass M of the central body is fixed for a given system, which is why only r and v trade off.

What earns the marks

• States that gravity provides the centripetal force keeping a satellite in a stable orbit.
• Explains that a higher orbital speed requires a smaller orbital radius for a stable orbit.
• Explains that a lower orbital speed requires a larger orbital radius for a stable orbit.
• Recognises that the mass of the central body is fixed, so speed and radius are inversely related for stable orbits.
• Rejects the idea that a satellite can change speed while keeping the same radius and remain in a stable circular orbit.

Common mistakes and how to put them right

• Assumes a faster satellite stays at the same radius.

  Put it right by stating that a stable orbit at higher speed needs a smaller radius.

• Claims that a larger radius gives a higher orbital speed.

  Put it right by stating that a larger radius gives a lower orbital speed.

• Treats the satellite's own mass as the quantity that sets the orbital speed.

  Put it right by noting that the central body's mass sets the relationship between speed and radius.

Exam tips

• Link the answer explicitly to gravity acting as the centripetal force rather than describing orbits loosely.
• Use the phrase 'for a stable orbit' to show you understand the condition being applied.
• State the direction of the change clearly: increased speed pairs with decreased radius, and decreased speed pairs with increased radius.