# Math Problem: How Long Until the Earth Falls Into the Sun?

Our earth has orbital motion, revolving once around the sun in about 365 days. Suppose that this orbital motion suddenly stopped completely, but everything else remained the same. How long would it take for the earth to plunge along a straight line into the sun?

[...]

Kepler's law applies to planetary orbits, whether they be of circular, or elliptical shape. It says that * T_{2}^{2}/T_{1}^{2} = R_{2}^{3}/R_{1}^{3}*, where

**is the period of an orbit and**

*T***is its semi-major axis. The semi-major axis is the average of the planet's maximum and minimum distances from the sun.**

*R*Let the earth's mean radius be * R_{1}*. Now, if the earth's orbital momentum were suddenly reduced (without exerting anything but a tangential stopping force on the earth), it would fall straight to the sun. This straight fall can be considered 1/2 of a degenerate elliptical orbit with major axis equal to

*. Its semi-major axis is*

**R**_{1}*(the average of*

**R**_{1}/2**and zero). Its period will be designated**

*R*_{1}*.*

**T**_{2}So: **T _{2}^{2}/T_{1}^{2} = (R_{1}/2)^{3}/R_{1}^{3} = (1/2)^{3}**

And therefore, * T_{2} = T_{1}/2^{3/2} = 0.353 year*, and the time to fall into the sun is 1/2 of that, or 0.176 years or 64.52 days—a bit over two months.

## Notes:

**Folksonomies:** mathematics astronomy problems

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/business and industrial/energy/renewable energy/solar energy (0.297890)

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**Concepts:**

Kepler\'s laws of planetary motion (0.977474): dbpedia | freebase | yago

Planet (0.881518): dbpedia | freebase | opencyc

Orbit (0.868543): dbpedia | freebase

Astronomical unit (0.689006): dbpedia | freebase | yago

Moon (0.603325): dbpedia | freebase

Elliptic orbit (0.583154): dbpedia | freebase | yago

Semi-major axis (0.582198): dbpedia | freebase | yago

Celestial mechanics (0.577764): dbpedia | freebase

**Physics Problems to Challenge Understanding, emphasizing concepts, and insight.**

**Electronic/World Wide Web>**

**Internet Article:**Simanek, Donald

*, Physics Problems to Challenge Understanding, emphasizing concepts, and insight.*, Retrieved on 2015-04-27

**Folksonomies:**games mathematics