how can we get distance of stars by spm ?

Answer 1

What do you mean by SPM? For me, the only SMP I know related to astronomy is the "Southern Proper Motion" program (Yale University and Universidad Nacional de San Juan, Argentina).

In order to find the absolute proper motion of individual stars (i.e., their true motion in relation to the Galaxy), they measure the apparent proper motion (in arc-seconds per year) and they use the star's parallax to find its distance (the parallax is found just like Leonardo D describes).

Because Yale University is a major partner, I suspect they use The General Catalogue of Trigonometric Parallaxes, Fourth Edition, 1995; by W. F. van Altena, J. T. Lee and E. D. Hoffleit,
(Yale University Observatory)

Imagine a triangle with the Sun, the Earth and the star (other than the Sun, of course). To find the very small angle at the star, take the star's position at 6 month interval and divide by two. This will be a very small angle (it is less than one second of arc, even for the closest stars). The base of the triangle is the Sun-Earth distance (1 Astronomical Unit).

One unit used is called the parsec (parallax second) and is the inverse of the angle. In other words, if the parallax angle is 0.2 seconds, then the distance is 5 parsecs.

1 parsec is 3.26 light-years or 206,265 AU.

Answer 2

You measure the star's position in the sky from two different locations - Best to do it when the earth is at opposite sides of its orbit. From the change in the measured position, you can calculate with simple trigonometry the star's distance.