How can I calculate distances between stars? (Advanced)

I am in 10th grade and consider myself a fairly competent amateur astronomer. I am doing a project on distribution of certain objects using some of the newish redshift data, I have been looking and looking for a formula that I could use to change RA, DEC and distance to X,Y and Z. I want to do this so that i can measure distances from one object to the next which I can't do unless i put the objects on a coordinate grid.

In mathematics and physics, we don't only use cartesian coordinates (x,y, and z). For some kind of problems we use other kinds of coordinate systems. In your case, the natural coordinate system to use would be 'spherical coordinates'. You can look at the web page:, and I will tell you how to relate these coordinates to your problem. (by the way, if you don't know this webpage yet, it is a reliable source of information about math).

In spherical coordinates, you have 3 coordinates, just like you have x, y and z in cartesian coordinates. These coordinates are: r, which is the distance between the origin and a point, and then two angles that work just like latitude and longitude on Earth, or as RA and DEC in the sky.

So if you want to apply these to your objects you have to set r as the distance to the object (which you get from its redshift, as you know). Then 'theta' is the azimuthal angle, which means it is your RA. But watch out! RA is generally given in units of hours, minutes, and seconds, which you will have to convert to decimal degrees. And finally, the 'phi' angle is your DEC. But once again, if your declinations are in degrees, minutes and seconds, you will need to convert them to decimal degrees. DEC goes from 90 degrees at the north celestial pole to -90 at the south celestial pole, but you need to have 'phi' from 0 to 180 degrees, so you should do " phi=90degrees-DEC ".

Now all your objects should be described by these coordinates. But to measure distances between them, the easiest for you might be to convert these coordinates to standard x,y, and z coordinates, and then calculate distance as usual. If you go to the web page I told you about before, you will get formulas to go from r, theta and phi to x, y, and z (equations 4, 5, and 6 as they number them). This should do the trick for you!

Good luck with your project!


This page updated on June 27, 2015

About the Author

Amelie Saintonge

Amelie is working on ways to detect the signals of galaxies from radio maps.

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