# Newton's Equation

1. Apr 4, 2005

### ammann

I have a question regarding a simple application of Newton's equation of gravity. As an exercise, I applied the equation to the EARTH - MOON - SUN system to see the relative gravitational force of the SUN and the EARTH on the MOON. I found something very interesting.

I chose the following conditions: New MOON at apogee; EARTH at aphelion. According to the Handbook of

Chemistry and Physics the data was as follows:
SUN EARTH MOON
Distance from Earth (km) 1.520 E08 --------- 4.055 E05
Mass (kg) 1.991 E30 5.979 E24 --------*

From this I found the distance from the SUN to the MOON to be 1.498 E08 by subtracting the EARTH - MOON distance from the EARTH - SUN distance.

Since I was interested only in the relative forces of the EARTH and the SUN on the MOON I could simplify the gravitational formula by eliminating the gravitational constant and the mass of the MOON. Newton's full formula is F = G x M1 X M2 / (D x D). The formula for relative force then is F = M / (D x D).

The following are the results:
SUN force on the MOON = 8.872 E13
EARTH force on the MOON = 3.636 E13
Resulting in a relative force of SUN / EARTH as 2.44.

I found that the SUN's force was almost 2-1/2 times stronger than the EARTH's. This, of course, is ... IMPOSSIBLE! If the MOON is to maintain its orbit around the EARTH, not only must the EARTH's force be consistently stronger to counteract the SUN's pull, but it must also be strong enough to act as a tether on the MOON and re-curve it toward the EARTH (away from the SUN), especially when it is at apogee. Only when the SUN's force is less than the EARTH's should the MOON begin to re-curve back toward the EARTH.

Is there something wrong with Newton's equation? Am I missing something obvious? Is there some other force that acts to compensate for the SUN's stronger gravitational attraction for the MOON?

Last edited: Apr 4, 2005
2. Apr 4, 2005

### dextercioby

Both Earth & Moon revolve around the Sun,so there's no surprise about it...

Daniel.

3. Apr 4, 2005

### tony873004

The Sun does pull the moon more than twice as hard as the Earth does. However, the Sun also pulls the Earth with almost the same strength that it pulls the moon. The Earth is getting pulled with the Moon, and that tends to negate the Sun's influence as far as the Earth/Moon system is concerned.

The difference in how hard the Sun pulls the Earth and how hard the Sun pulls the Moon is called the Solar gravitational tide. It is what is important, not simply how hard the Sun pulls the Moon. The solar gravitational tide does have an influence. It is why the Moon's orbit is elliptical instead of round, and it is why the Moon's orbital nodes rotate in an 18 year cycle.

4. Apr 4, 2005

### pervect

Staff Emeritus
Nope. It's about right, IIRC. That's why, for instance, Isaac Asimov calls the Earth-Moon system a "double planet". Think of it as a pair of bodies orbiting around the common center of mass, not just as a satellite orbiting a primary.

The Moon is still in the Earth's "Hill sphere", so it can't escape at the current time. The detailed analysis is a bit mathematical, but for a brief discussion see for instance

http://en.wikipedia.org/wiki/Hill_sphere

You can also get more detail by searching the forum for the phrase "Hill sphere" or "Hill radius".

5. Apr 4, 2005

### Janus

Staff Emeritus
A little addendum to this is that if you plot the Moon's heliocentric orbit, (its path around the Sun), even though it weaves in and out from the Sun, the curve of this orbit is concave to the Sun at all points.