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 T₂²/T₁² = R₂³/R₁³, where T is the period of an orbit and R is its semi-major axis. The semi-major axis is the average of the planet's maximum and minimum distances from the sun.

Let the earth's mean radius be R₁. 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 R₁. Its semi-major axis is R₁/2 (the average ofR₁ and zero). Its period will be designated T₂.

So: T₂²/T₁² = (R₁/2)³/R₁³ = (1/2)³

And therefore, T₂ = T₁/2^3/2 = 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.

Donald Simanek, Physics Problems to Challenge Understanding, emphasizing concepts, and insight.

• astronomy
• mathematics
• problems