Earth, Jupiter, and Uranus all revolve around the sun. Earth takes 1 yr, Jupiter 12 yr, Saturn 30 yr, and Uranus 84 yr to make a complete revolution. One night, you look at those three distant planets and wonder how many years it will take before they have the same position again.
a) How often will Jupiter and Saturn appear in the same position in the night sky as seen from Earth tonight?
b) How often will Jupiter, Saturn, and Uranus appear in the same direction in the night sky as seen from Earth?
660 years
660 years
To answer both questions, we need to find the least common multiple (LCM) of the revolution periods for the planets involved. The LCM will give us the amount of time it takes for all three planets to align again.
a) To find how often Jupiter and Saturn appear in the same position in the night sky tonight, we need to find the LCM of their revolution periods, which are 12 years and 30 years.
To calculate the LCM, you can use the prime factorization method or the method of finding multiples:
Prime factorization method:
- Write the prime factorization of each number: 12 = 2^2 * 3 and 30 = 2 * 3 * 5.
- Take the highest power of each prime factor that appears in either factorization: 2^2 * 3 * 5 = 60.
So, it takes 60 years for Jupiter and Saturn to appear in the same position in the night sky again.
b) To find how often Jupiter, Saturn, and Uranus appear in the same direction in the night sky tonight, we need to find the LCM of their revolution periods, which are 12 years, 30 years, and 84 years.
Using the prime factorization method:
- Write the prime factorization of each number: 12 = 2^2 * 3, 30 = 2 * 3 * 5, and 84 = 2^2 * 3 * 7.
- Take the highest power of each prime factor that appears in either factorization: 2^2 * 3 * 5 * 7 = 420.
So, it takes 420 years for Jupiter, Saturn, and Uranus to appear in the same direction in the night sky again.