Suppose you received a message from aliens living on a planet orbiting a star identical to our Sun. They say they live 4 times farther from tier star than Earth is from the sun. What is the length of their year compare to ours?
For all planets orbiting the sun, or another star of the same mass as the sun,
P^2*/R^3 = constant
P is the period and R is the radial distance.
This is called called Kepler's third law. http://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion#Third_law
Since R^3 is 64 times larger for the other star's planet (in your problem), P^2 must be 64 times larger also. The Period ratio is the square root of 64.
is 8 the correct answer
To determine the length of a year on the alien planet, we first need to know the length of Earth's year.
Earth orbits the Sun at an average distance of about 93 million miles (or 150 million kilometers). It takes approximately 365.25 days for Earth to complete one full orbit around the Sun. Therefore, we can say that Earth's year is approximately 365.25 days long.
The aliens mentioned that they live 4 times farther from their star than Earth is from the Sun. So, to find their distance from their star, we can multiply Earth's average distance from the Sun by 4.
Distance of the aliens from their star = 4 * 93 million miles (or 150 million kilometers).
Once we have the distance of the aliens from their star, we can use the fact that the time it takes for an object to complete one orbit is directly proportional to the distance from the object it is orbiting.
In this case, we have the distance of the aliens from their star (which is 4 times larger than Earth's distance from the Sun), so we can conclude that the length of their year will be 4 times longer than Earth's year.
Length of the alien year = 4 * 365.25 days.
Therefore, the length of the alien year would be approximately 1,461 days, assuming that the alien planet's orbit is similar to Earth's.