How many times each year are earth, Jupiter, and the sun all lined up?
It takes Jupiter 11.86 years to orbit the sun. How many times each year are earth, Jupiter, and the sun all lined up? Is it the same number each year or does it vary from year to year? Explain.
Jupiter takes 11.86 Earth years to orbit the sun (the Earth takes one). Therefore Jupiter orbits the sun 11.86 times faster than Earth.
Each minute (1 degree of 360 degrees) of Earth's rotation, Jupiter travels 360/11.86 = 30.35413 degrees round the sun.
We want to find when
`t -= 30.35413t` (mod 360)
t = 12, 30.35413t = 4.2496 (mod 360)
t = 13, 30.35413t = 34.6037 (mod 360)
Therefore the two planets have been in alignment between t = 12 and 13 degrees of the Earth's rotation.
t = 24, 30.35413t = 8.4992 (mod 360)
t = 25, 30.35413t = 38.8533 (mod 360)
Therefore the two planets have been in alignment between t = 24 and 25, ie t = 2 x 12 and 2 x 12 + 1.
Similarly, they will be in alignment between 36 and 37, 48 and 49, 60 and 67, ... 348 and 349. That is, eleven times in an Earth year. This is equal to the number of times faster Jupiter's orbit is than Earth's, rounded down to the nearest integer. They will be in alignment 11 times every year, but a different points in the year, but always 30.35 degrees apart around the circle of orbit.
They are in alignment 11 times (11.86 rounded down to the nearest integer) every (Earth) year since Jupiter's year is 11.86 times shorter than Earth's. This is true every year, but they align at different times throughout the year.