<p>This is not really an astronomy so much a basic Maths problem.</p>

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

<p><span class="math-container">$$\vec{v} = v_x \hat{i} + v_y \hat{j} + v_z \hat{k},$$</span>
where these components are the two proper motion velocities and the radial
 velocity.</p>

<p>Assume no acceleration, so that
<span class="math-container">$$\vec{r} = \vec{r_0} + \vec{v} t $$</span></p>

<p>Use Pythagoras to get the distance to the Earth using the components of <span class="math-container">$\vec{r}$</span> at any time <span class="math-container">$t$</span> and then find the time at which this is a minimum (and if you are unsure how to do that <span class="math-container">$\rightarrow$</span> Maths SE).</p>

CC BY-SA 4.0
2018-11-04T12:56:54.357Z
2018-11-04T19:38:24.057Z
2018-11-04T19:38:24.057Z
ProfRob
<p>How should I approach this problem?</p>

<ol>
<li>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.</li>
<li>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?</li>
</ol>

<p>I know how to do 1. but I am not sure how to approach 2.</p>

<distances><declination><right-ascension>
How to calculate the time and distance of closest approach of a star?
AHMED KRS
# How to calculate the time and distance of closest approach of a star?

(Asked by: username_0 on 2018-11-04)

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.

**Tags:** distances, declination, right-ascension


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## 1 answers

### Answer by username_1 on 2018-11-04. Score: 1 | Accepted answer

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).


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