Question

How can we find the distance to a star that is too distant to have a measurable parallax?

Answer 1

If m is visual magnitude of a star, M is the absolute magnitude and d its distance from us, `M=m-5 log (d/10)`.

Explanation:

Brightness faints in proportion to square of the distance.

Brightness is measured from light from the star.

With the notations m for visual magnitude, M for absolute magnitude and d for the distance of the star from the observer,
the distance is `10^s` parsec, where s is given by

s = 1 + 0.2 (m - M).

This could be modified as

`M = m - 5 log (d/10)`,

using `d = 10^s, and so, s = log d.`.

Comparison with other stars is also useful. The formula used is

`M_1-M_2=100^(1/5)(L_1/L_2)`, L being luminosity, with L = 1 for Sun.

For comparison with Sun, `M_1=4.83, m_1=-26.74 and L_1=1`.

Pogson's ratio `100^(1/5)=2.51193`, nearly.

Madras ( now Chennai) based Pogson/s research in 19th century is quite relevant..

Reference: http://window2universe.org/kids_space_stars_dist.html.

I think that I have paved the way for further studies, by the interested readers. .

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