system
Given the following post, respond as if you're commenting on Reddit:
human
How does Newton's Law of Gravity help us predict the orbits of the planets with "great accuracy"? (Please read description.)

I'm reading *A Brief History of Time* by Stephen Hawking. On pg. 17, there is a paragraph that reads:

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"Newton's law of gravity also tells us that the farther apart the bodies, the smaller the force. Newton's law of gravity says that the gravitational attraction of a star is exactly one quarter that of a similar star at half the distance. This law predicts the orbits of the earth, the moon, and the planets with great accuracy. **If the law were that the gravitational attraction of a star went down faster with distance, the orbits of the planets would not be elliptical, they would spiral in to the sun. If it went down slower, the gravitational forces from the distant stars would dominate over that from the earth.**"

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What does the part in bold mean exactly? (As is clear, I'm a layperson, so please keep it simple.) Thank you!
gpt
Newton's law of gravity tells us that the force of gravity drops with the square of distance. So if you are twice as far away from the Sun, you feel one quarter as much gravity. We call this an "inverse square law", because the mathematical equation 1/r^2 is called an "inverse square".

It turns out that an inverse square law has a lot of very convenient mathematical properties.

Firstly, if the gravity somewhere is dominated by a single object, then you get very simple orbits. This is called a "Keplerian" system. Our Solar System works like this, because the Sun dominates the gravity by far - it has over 99% of the mass of the Solar System.

In a Keplerian system, orbits always perfectly repeat themselves. If you have an elliptical orbit, you're sometimes closer to the Sun, and sometimes further from the Sun. In a Keplerian system, these points always happen at the same stage in your orbit. So you are always closest to the Sun at one time of year, and you are always far from the Sun at one another time of year. This consistency makes it really easy to predict orbits over a long period of time.

This is a lot more complicated if you have a lot of different objects producing the gravity. For example, in a galaxy you have lots of gas and stars and dark matter producing gravity together. Even though each of these has an "inverse square law" for gravity, all of these sources of gravity add up to give us a different law for the galaxy's gravity. This makes it a lot harder to predict the position of stars around a galaxy over time. The maths doesn't work out so simply. Stars can orbit in a "spirograph" kind of pattern where they are closest to the centre several times in an orbit. They bob up and down too. They don't follow a simple ellipse around the Milky Way.

Secondly, there is a maximum amount of acceleration that you can get from an inverse square law. This is difficult to grasp without calculus. The basic idea is that if you were falling onto a star from your location, the speed you'll reach when you hit its surface is given by adding up all the acceleration you feel as you fall onto it. If you flip this, this also gives you the speed you would need to escape from the star's surface back to that point.

With an inverse square law, the speed you need to escape to a certain distance from a star constantly increases. *But* it reaches a maximum. This is the escape velocity. Above the escape velocity, the force of a star will never add up to enough to pull you back in again.

However, if the force of gravity drops down just a little bit slower than an inverse square law, then this speed never reaches a maximum. That means there is no escape velocity, and you can never completely escape from a star. No matter how fast you are going, a distant star will *eventually* pull you back.

This means that we can't just think of the Solar System as an isolated system. We would need to consider the effects of every other star in the galaxy, because we haven't escaped them. This would make orbits extremely difficult to calculate, and probably wouldn't let you have the nice simple hierarchy of orbits that we have at all.