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How to measure the distance of a star and the diameter of it from earth based tools? Normally we can measure the angle between the stars edge. But diameter or distance needed to measure the other parameter. So how the scientist measuring this both parameter?

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To measure the size and distance of stars you use a set of tools that build on each other.

For distance, first there is parallax. Nearby stars have an apparent shift in position relative to distant stars or preferably, galaxies, in the 6 months it takes the Earth to go from one side of the Sun to the other. Triangulation gives the distance.

Then a table of luminance based on star type (spectrum) or behavior (Cepheid and RR Lyre variables which are amazingly consistent from star to star) can be used for stars further away. The variable stars can even be seen in nearby galaxies and their brightness gives a good measure of their distance.

Knowing the distance from stellar brightness lets you make the next jump to using redshift. You can see the small redshift in local galaxies to calibrate to the current popular value for the Hubble Constant and measurements from there on use redshift. I think some luminance of quasars is used for really great distances, but I am not up to date on that.

For size, stars like the Sun can be used as a reference and some nearby stars were measured with interferometric methods developed (and first performed) by Michelson. Radio astronomy using long baseline interferometry gives very good size info since it depends on coherence length, and therefore size of the source. Elaborations and variations on this have produced size info on stars all over the H-R diagram of stellar mass and luminance.

The work by astrophysics on energy produced in stars and their size versus mass versus spectra has been outstanding and I'm over my head here, but I think those results are of primary use today in determining size. Basically you can give someone the spectra and they can tell you the mass and size of a star.

When we get arrays of optical telescopes in space, direct measurement will be possible on much more distant objects than have been measured so far and this will all get refined by an order of magnitude (or several orders of magnitudes since even planets around stars can be imaged). Note that by "direct measurement" with arrays I really mean optical measurements of something that after a great deal of computation produces an image.

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