Question

the nearest star outside our solar system, proxima centauri, is about 4.2 light years away. from earth we have measured its parallax angle to be 0.77 arcsecond. if somehow the earth were to orbit the sun at twice its current orbital distance, what parallax angle would we observe then (with small angle approximation)?

Answer 1

Upon doubling the Earth's orbital distance to 2 AU, the observed parallax angle for Proxima Centauri would be 0.385 arcseconds, half of the currently measured 0.77 arcseconds at 1 AU.

Step-by-step explanation:

If the Earth's orbit around the Sun were doubled, the new orbital distance of the Earth would be two astronomical units (2 AU). The parallax angle, which is inversely proportional to the distance, would decrease by a factor of two. This is because orbital distance and parallax angle have an inverse relationship as per the small angle approximation—which is valid for very small angles like the ones involved in astronomical observations.

Currently, Proxima Centauri has a parallax angle of 0.77 arcseconds at 1 AU. Double the distance to 2 AU implies the parallax angle would be halved. Thus, the new angle would be 0.385 arcseconds.

This also corresponds with the concept related to parsecs, where a parsec defines the distance to a star whose yearly parallax is 1", and with 1 parsec equal to 3.26 light years. Even with a doubled Earth orbit, Proxima Centauri would remain significantly farther than 1 parsec.