# How to calculate the various properties of a star?

I am wanting to plot a velocity profile (a plot of velocity against the star's distance from the galactic core) from the Gaia DR2, but so far my efforts have been disappointing.

The information I have for each star are the declination ($\delta$), right ascension ($\alpha$), radial velocity ($v_r$), parallax ($\theta_\text{parallax}$) and proper motion ($\mu$) (ra and dec).

I tried to calculate the star's distance from the galactic core using the cosine rule:

$$d = \frac{1000}{\theta_{\text{parallax}}}$$ $$\theta = \arccos{[\sin{\delta_\text{star}}\sin{\delta_\text{core}} + \cos{\delta_\text{star}}\cos{\delta_\text{core}}\cos{(\alpha_\text{star}-\alpha_\text{core})}]}$$ $$r = \sqrt{d_\text{core}^2+ d^2 - 2d \times d_\text{core} \times \cos{\theta}}$$

Where $d$ is the distance from the Sun to the star. $d_\text{core}$ is the distance from the Sun to the galactic core. $\theta$ is the angular separation of the star and the galactic core. $r$ is the distance from the star to the core.

And the star's speed using the equations:

$$\mu = \sqrt{\mu_\text{dec}^2 + \mu_\text{ra}^2}$$ $$v_t = 4.7 \mu d$$ $$v = \sqrt{v_r^2 + v_t^2}$$

But the plot doesn't show any kind of trend (just a kind of blob). Have I calculated correctly and, if not, how could I calculate the tangential velocities and radius from the galactic core?

Basically, I want to make the kind of plot that helps us identify dark matter in galaxies.

• Note that I am placing restrictions on the allowed parallax error, as advised Commented May 21, 2018 at 19:36
• You haven't calculated a velocity, you have calculated a speed. You need the velocity at a tangent to the line from the star to the Galactic centre. Commented May 21, 2018 at 23:00
• Also note that given a 0.1 mas systematic error in parallax, you won't get accurate tangential velocities beyond about 2kpc from the Sun. Commented May 21, 2018 at 23:02
• @RobJeffries Would that be standard for a telescope like Gaia? It seems like a large uncertainty to me Commented May 22, 2018 at 6:14
• @B--rian I haven't unfortunately, but I might revisit it over the summer Commented Jun 29, 2021 at 13:17