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Once, I was staring at the sky and wanted to know the distance to stars I could see. I searched the Internet but didn't find any easy to use tools. The distance to stars can be measured using sophisticated instruments. But are there any simple ways to measure the distance of a star approximately, using the least instruments possible? Accurate distance is not necessary, but rather an approximate value to get a feel for the distance.

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If you are patient and the star is close enough you can use parallax. The idea is that close objects shift positions with respect to the background when the observer moves. A very intuitive example is: extend your arm and look at your thumb. Now close one eye at the time and see how the position of the thumb changes! You can actually calculate the distance from your eye to your thumb my measuring how much it shifts and the distance between your eyes, it is very simple (see below)

The caveat here is that the longer your arm, the smaller the shift against the background. So for a star closing an eye of the time is not gonna cut it, instead you have to move a large distance to see it shift around. The longer distance you can move is by waiting six months (hence the 'be patient' part).

If you know the diameter of Earth's orbit (the equivalent to the distance between your eyes in the example above) and how much the star shifts w.r.t to the background (angle, $\theta$) stars then you can figure out its distance

$$ d \approx \frac{{\rm 1} \text{AU}}{\theta} $$ Where 1 AU (149,600,000 km) is the average distance from the Sun to the Earth.

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  • $\begingroup$ This cannot be done (for stars) without sophisticated instruments. $\endgroup$
    – ProfRob
    Sep 28, 2018 at 11:33
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There are simple methods to compute distances between stars (the simplest being simply the parallax method), but none will work without sophisticated instruments. That is why nobody managed to compute the distance between stars until the 19th century, as we did not yet have instruments precise enough to perform such a calculation.

What you can do relatively simply is compute the distance of the moon. As this was performed in $\approx 300\ \text{BCE}$, you do not need any too complicated instruments. There are two methods for this, the first one being the method of Aristarchus, by using various relations from the shadow of earth projected on the moon and angles between the sun and moon (more details here for instance), and the second is the parallax method used directly, as used by Hipparchus, although this one takes a bit more coordination.

The Hipparchus methos is the following : look at a solar eclipse in two different locations $A$ and $B$. If you note that, at the very same moment, the moon completely covers the sun in $A$ while it only covers a fraction of it at $B$, then you can compute the angle formed by those two locations and the sun, as a function of the sun's angular size and this fraction.

From there, it is simply a matter of trigonometry, since you have the distance between the two locations $A$ and $B$ and the angle $\alpha$ at the top of this triangle (since the distances involved are very long with a very small basis, we can assume that both sides of the triangle are roughly equally the distance to the moon).

Of course, this requires the ability to see the event in both location (either by your ancient greek friends in other locations or a webcam).

If you wish to perform the same measurement for stars, I'm afraid it will be much harder. The parallax trick will also work, by measuring angular coordinates of the star 6 months apart (so that the earth is at its two different locations $A$ and $B$), but it may not work unless you have rather good equipment.

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    $\begingroup$ Astronomers were making reasonably precise calculations of the distance to nearby stars before the 19th century, but they could only express those distances as multiples of the astronomical unit---the radius of Earth's orbit around the Sun. The big breakthrough in the mid-19th century was not a sudden leap in the quality of their instruments: it was the invention of a way to accurately relate the astronomical unit to a distance measured on the face of the Earth. en.wikipedia.org/wiki/Transit_of_Venus $\endgroup$ Mar 12, 2018 at 16:32

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