By NASA.Mrshaba at en.wikipedia [Public domain], from Wikimedia Commons

If the earth stopped in its orbit, how long would it take to fall into the sun?

Short answer: About 65 days.

Long answer: What would happen if the earth came to a dead stop in its orbit? Don’t worry, this can’t actually happen. Orbits aren’t nearly as sensitive as Hollywood would have you believe – the earth has been going around the sun for almost 5 billion years now without any serious disruption, and it’s not going to stop any time soon.

But what if it did? In much the same way that a ball released at the top of a tower will accelerate towards the ground, the earth will begin to accelerate towards the sun. Let’s calculate the time to impact.

The direct approach is difficult. Since the earth is getting closer to the sun the force of gravity between them is getting stronger so the earth is going to accelerate faster, which involves integrating the force of gravity on a computer. I’m lazy, so I won’t do that.

Instead, I’m going to remember Kepler’s laws. Planet’s orbits are ellipses with the Sun at one focus. Presently, the earths orbit around the sun is very close to circular, with a radius of 1 AU (that’s about 150 million km). Throughout the year it doesn’t tend to get too much closer or too much farther from the sun. Instead, if we imagine the earth with a highly elliptical orbit, we see a different picture. Bodies in elliptical orbits are slow at the most distant point in the orbit but they fall faster and faster as they approach, eventually whipping around the back side and getting launched back to where they started.

By Brandir [GFDL (http://www.gnu.org/copyleft/fdl.html), CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/) or CC BY-SA 2.5 es (http://creativecommons.org/licenses/by-sa/2.5/es/deed.en)], via Wikimedia Commons

Why does this matter? Well, our earth (stopped dead in its tracks) is now on an elliptical orbit! It’s a very specific elliptical orbit where the ellipse has been collapsed to a line which passes through the sun, but it’s an elliptical orbit nonetheless! The ‘radius’ of this orbit is half of the long direction of the ellipse, called the semi-major axis, which would be half of the current distance to the sun.

Since we know the semi-major axis, we can use Kepler’s third law – it relates the semi-major axis of the orbit to the orbital period:

KeplersThird

where P is the orbital period, a is the semi major axis, G is the gravitational constant, and M is the mass of the sun.

Plugging in 0.5 AU for the semi-major axis, we find that the orbital period is 129 days [1]. This number is the amount of time required to complete one full orbit, so half of this number would tell us how long it takes to travel half the orbit – from the earth to the sun. Thus, it takes about 65 days for the earth to hit the sun.

 

 


 

 

asked by /u/Echrome
image credit: Wikimedia Commons

 


 

 

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