Show all the work for the following 1. The distance to Alpha Centauri is 4.26 li
ID: 1894536 • Letter: S
Question
Show all the work for the following1. The distance to Alpha Centauri is 4.26 lightyears. When Earth and Mars are nearest each other in their orbits, Mars is 25” (arcseconds) in diameter on the sky. What is the ratio of the size of Alpha Centauri’s parallax (as seen from Earth) compared to the apparent size of Mars at Earth’s closest approach?
2. On Earth it takes 6 months to measure the parallax of a star. Why? How much longer (in months) would it take to measure a star’s parallax from Mars?
3. Using the general parallax equation D = d/p, by what percentage would the parallax of ANY star change if we were to observe it from Mars instead of the Earth? Is this percentage larger or smaller than what is seen from Earth.
4. What is the parallax of Alpha Centauri (in arcseconds) as seen from Mars?
Explanation / Answer
Alpha Centauri is 4.26 ly = (4.26 ly)(9.461 * 1015 m/ly) = 4.030 * 1016 m away.
The earth's orbit is 2.992 * 1011 m across. So the parallax of Alpha Centauri is (2.992 * 1011)/(4.030 * 1016) = 7.424 * 10-6 radians.
25 arcseconds = (25 arcseconds)(4.848 * 10-6 radians/arcsecond)= 1.212 * 10-4 radians
So the ratio of Alpha Centauri's parallax to Mars' apparent size is (7.424 * 10-6)/(1.212 * 10-4) = 0.06125 or about 1/16. That is the answer to (1).
(2) It takes six months because you measure from opposite sides of the Earth's orbit, which takes half a year. On mars it would take 343.48 days to travel to the opposite side of its orbit, or about 11.3 months, so the answer to (2) is 5.3 months.
(3) The parallax would increase because Mars' orbit is larger. The increase is directly proportional to the size of the orbit, or (4.559 * 1011 m)/(2.992 * 1011 m) = 1.524. So the answer is 52.4%.
(4) The parallax from Mars is (4.559 * 1011)/(4.030 * 1016) = 1.131 * 10-5 radians. Which is 2.33 arc seconds.