Gravitational Tides

Look closely at the gravitational force acting on a moon as it orbits its planet:

If we subtract the center of mass force, we see the differential force acting on it:

So gravity "stretches" and "squashes" a moon!

Let's look at this mathematically. The force of gravity is:

So the differential force (also called the tidal force) across a distance dr is
Note that
• the tidal force is proportional to the mass of the primary (M)
• the tidal force is inversely proportional to the distance cubed.
Note also that it works both ways -- the moon also stretches the planet!

Why is it called a tidal force?

What is stronger on the Earth, the tidal force from the moon or the tidal force from the Sun?

So the moon exerts a stronger force, but the Sun's tidal force can be significant. Hence the concept of spring tides and neap tides:
• Spring Tides: Sun and Moon in alignment; tidal forces add. Big tides!
• Neap Tides: Sun and Moon 90 degrees apart; tidal forces counteract. Small tides.

Remember: Tides are not merely a water effect! The Earth's surface also has tidal bulges, about 10cm in height. And the moon has an even greater tidal bulge -- 20m high.

Thought experiment: What happens when you keep squeezing and stretching a piece of silly putty? What does this have to do with tides?