How to calculate the time and distance of closest approach of a star?

Question

How should I approach this problem?

1. A star has a motion on the sky of 0.78 arcsec/yr in declination and 1.33 arcsec/yr in right ascension. Calculate the velocity of the star in the plane of the sky in km/sec.
2. The star also has a radial component of its velocity, coming towards earth at 4.3 km/s. What is the time and distance of closest approach?

I know how to do 1. but I am not sure how to approach 2.

Answer 1

This is not really an astronomy so much a basic Maths problem.

Put the Earth at the origin and the star at $\vec{r}(t)$ with a position $\vec{r_0}$ at $t=0$. Define your axes such that the star is in the z direction and its tangential motion components are along the x and y axes.

$$\vec{v} = v_x \hat{i} + v_y \hat{j} + v_z \hat{k},$$ where these components are the two proper motion velocities and the radial velocity.

Assume no acceleration, so that $$\vec{r} = \vec{r_0} + \vec{v} t $$

Use Pythagoras to get the distance to the Earth using the components of $\vec{r}$ at any time $t$ and then find the time at which this is a minimum (and if you are unsure how to do that $\rightarrow$ Maths SE).