# Units in distance parallax relation

Relation between distance d and paralax plx is:

$$d (pc) = 1 / plx(arc)$$

When I want to express plx in mas, will it be

$$d (pc) = 1/[plx(mas)*1000]$$

or $$d (pc) = 1000/plx(mas)$$

Thank you very much

To convert mas to arcseconds, you divide by 1000, so the proper equation would be: $$d(pc)=\frac{1}{plx(mas)/1000}$$

Which can be rewritten as your second equation: $$d(pc)=\frac{1000}{plx(mas)}$$

I'll add that the above is an approximation algorithm, the true relation between distance and parallax is:

$$d = \frac{1au}{\tan \theta}$$ d = Distance in AU

$$\theta$$ = Angle in degrees ($$\theta = p/3600$$)

On any modern processor, there is little need to ever use the approximation.

• If we want to be pedantic: The nearest star has a parallax of 0.76813 arcsecs. For this star, $\theta/\tan\theta \simeq 1 - 5\times 10^{-12}$. The correction is smaller for more distant stars and is in any case smaller than uncertainties in the measured parallax and smaller than the error in inferred distance introduced by assuming the best-estimate distance is just the reciprocal of any measured parallax with an uncertainty (which it isn't). Commented Jul 1, 2022 at 16:59