Aristarchus's critics argued that parallax was not observed...

Aristarchus's critics argued that parallax was not observed and therefore the Earth could not be in motion. In fact we know now that the stars are very far away and parallax motion is very small. Let's examine this by estimating the change in angular size of the angular separation between two stars 10 light-years away from the Sun. The two stars are 3 light-years apart. Calculate the angular separation of the two stars when the Earth is closest to them in its orbit and when it is farthest away, on the opposite point on its orbit. If human vision (without the aid of a telescope) can at best distinguish a difference of about $0.02^{\circ},$ would the change in angular separation be detectable to the naked eye?

Key Concepts

Stellar Parallax

Stellar parallax is the apparent shift in position of nearby stars relative to distant background stars when the observer’s viewing position changes. This small angular displacement is critical in astronomy for determining stellar distances and demonstrates that, although Earth is in motion, the enormous distances involved make the observed parallax minute.

Angular Separation

Angular separation refers to the angle between two celestial objects as seen from the observer's perspective. It is a key concept in measuring and quantifying the apparent distance between stars, particularly when assessing changes due to varying positions of the observer as the Earth orbits the Sun.

Astronomical Distance Scale

Understanding the astronomical distance scale is vital when discussing parallax and angular measurements in astronomy. The immense distances to stars mean that even significant physical separations translate to very small angular separations, emphasizing why parallax effects are usually minuscule and challenging to detect.

Resolving Power of the Human Eye

The resolving power of the human eye defines the smallest angular difference that can be distinguished without optical aids. In this context, with a limit of about 0.02°, it highlights that the tiny changes in angular separation due to Earth's orbital motion are generally too subtle to be detected by the naked eye.

Transcript

00:01 Alright, this equation or question, we're looking at the parallax.

00:06 They define or explain the parallax to be involved in measuring a star.

00:13 So we have a point e1 here, another point e2, and then they have s over here.

00:24 Okay, so they define this angle that i've just drawn, right there.

00:29 That is called the parallax.

00:34 Let's say that in this situation, the parallax they found to be 0 .000 to 111 degrees.

00:45 Okay, they're asking us to estimate the distance of the star.

00:50 Okay, so this amount right here, take the distance from the earth to the sun to be 9 .3 times 10 to the 7.

01:01 So o represents the sun, right? if you recall, we have e, which you might assume stands for earth, but they said that they wait six months, right? the earth travels around the sun, one full revolution every 12 months.

01:18 So six months would put you right on the opposite side over here.

01:22 They're using this to calculate this huge distance so that they get some kind of perpendicular, like, this based on the angles reported from six months apart and they find them each of these parallaxes to be similar enough to well well truly in the opposite direction right they find that their trajectory is 90 degrees on the exact opposite side of the sun and that allows them to find this angle of this parallax angle that they can then use to find the distance between the sun over here and the star.

02:04 All right.

02:04 So they said the distance from the earth to the sun should be 9 .3 times 10 to the 7...

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