Planetary orbits and equilibrium

Tin Foil Hat

Warning - possible stupid question ahead!

My understanding of celestial orbits is that a small body will orbit a larger body, in space, when its gravitational attraction to that body is equal to its linear acceleration away from that body as it orbits.

My question - why is this system a stable equilibrium rather than an unstable equilinrium? E.g. why is the Earth not knocked from its orbit by the smallest meteorite strike, like blowing on a pencil balancing on its tip?

Hope this question makes sense.

FISMA

Tin Foil Hat wrote: »

My understanding of celestial orbits is that a small body will orbit a larger body, in space, when its gravitational attraction to that body is equal to its linear acceleration away from that body as it orbits.

Newton's First Law tells us that things do not accelerate, unless there is an unbalanced Force on them. Bodies in motion (with a constant velocity) tend to remain in motion (with a constant velocity) unless acted on by an unbalanced Force. Bodies at rest tend to stay at rest, unless acted on by an unbalanced Force.

In order for anything to go in a circle, there must be a Force that is not balanced.

This Force must act towards the center of the circle. We describe this Force by naming it by what it does. The Force that keeps you going in a circle is a center (centri) seeking (petal) - a centripetal Force - the Force acts inward.

There are many Forces that can be centripetal.

Tension is a centripetal Force when a rope keeps a ball going in a circle.

Friction is a centripetal Force when a car goes around a turn.

The Electric Force is a centripetal Force when an electron is accelerated, and so on.

A body in a circular orbit such as a satellite, is neither speeding up or slowing down, it is just changing its direction. The gravitational Force IS the centripetal Force that keeps the body going in a circle.

In reality, celestial bodies orbit around a center of mass of two or more objects.

For example, the moon does not simply go around the Earth, rather, the moon-Earth system orbit about a common center of mass, which is probably inside the Earth anyhow.

The whole point about an accelerating object is that there has to be an unbalanced Force. Thus, there should not be equal and opposite Forces.

Tin Foil Hat wrote: »

My question - why is this system a stable equilibrium rather than an unstable equilinrium? E.g. why is the Earth not knocked from its orbit by the smallest meteorite strike, like blowing on a pencil balancing on its tip?
Hope this question makes sense.

The Earth is incredibly massive and it takes a lot of Force to knock it from its orbit. I am willing to be that the recent earthquake in Japan has had an effect on the tilt of the Earth that we are able to measure.

<my snip>

The Earth may be in a stable orbit, however, it is not in stable equilibrium. Again, since it is accelerating, it cannot be in equilibrium - balanced Forces.

Stable equilibrium - topmost image. When a ball, initially at rest (in equilibrium), receives a slight displacement to the left or right, it will return to its original position via a restoring torque.

Unstable equilibrium - topmost image. When a ball, initially at rest (in equilibrium), receives a slight displacement to the left or right, it will not return to its original position. Instead a torque will cause it to become displaced further and more unstable.

Neutral equilibrium - bottom image. When a ball, initially at rest (in equilibrium), receives a slight displacement to the left or right, it will not return to its original position via a restoring torque. However, the new position will be just as stable as the original.

I take it from your post that you are questioning why the Earth is in a stable orbit.

Remember, it is the Sun-Earth system that is orbit about a center of mass (that is physically located inside of the Sun). It would take a lot of Force to deviate from this orbit.

The Earth has angular momentum and wishes to conserve this quantity. For small Forces, the Earth is more than able to stay in neutral eq or stable. It would take a lot of Force to cause unstable.

Hope this helped.

stoneill

The common center of orbit is called the barycenter.
And indeed for the Earth-Moon system it is inside the diameter of the Earth.

rccaulfield

It doesn't have to be equal, the earth would orbit the sun if it was twice the distance out albeit at a much slower rate. Look up Keplers laws of planetary gravitation that still hold today!
A meteor is a little bit of rock that burns up in the atmosphere so doesn't touch the planet but a large asteroid like the one 65 million years ago might minutely change the orbit by the smallest fraction, i don't think we can know for sure without surviving a hit!

Capt'n Midnight

if you look at the energies required to send a space probe to mercury (5 gravity assists en route) you should realise that it takes a lot of energy to move the earth to a slightly different orbit.