# Equation to find the speed of a distant star

1. Mar 6, 2009

### DeepSeeded

Does anyone know the equation used to find the speed of a distant star?

2. Mar 6, 2009

Staff Emeritus
There is no such equation. Two stars near each other can have quite different velocities.

3. Mar 6, 2009

### Ultrastar 1

That is not true. Such formulas do exist. Google it.

4. Mar 6, 2009

### DeepSeeded

The only equation I can find bases it on the redshift in the light, but speed is not determined soley by the red or blueshift recorded since these values change with distance also.

If you based the speed of star systems on shifts in wavelength you would be lead to beleive that the further away something is the faster it is moving.

5. Mar 6, 2009

### Nabeshin

No. Proper motion of a star is not in any way correlated with distance, so no formula could possibly exist. Cite what you're talking about, please.

Why do you say wavelength shifts change with distance? If you're referring to the hubble expansion, I'm not sure, but I don't think it applies for a gravitationally bound system such as the Milky Way, and thus, the constituent stars will not show any hubble redshift. (This may be incorrect, but even if it is the effect will be very small and easily compensated for knowing the distance of the star).

The Doppler shift gives very accurate velocity measurements, and assuming we can compensate for or ignore intervening dust between the star and us, is a good method for finding the velocity of a star. It's not without its faults though, because it cannot detect any kind of transverse motion.

6. Mar 7, 2009

### v2kkim

I like to ask what kind of speed you refer to. The recession speed which is universal space expansion or a local motion like fast moving star orbiting a massive black hole.

Last edited: Mar 7, 2009
7. Mar 7, 2009

### marcus

There are several methods. It might help if you provided some motivation, are you writing a paper? arguing with a friend? do you need examples of different methods?

But maybe I can just answer randomly without being sure what you want.

First-off cosmological redshift doesn't apply to stars in our own galaxy or nearby galaxies.
No distance expansion to worry about.

So any spectral shift you see indicates speed towards us or away from us.

For simplicity lets write the speed away from us as with a letter V as a fraction of the speed of light. If V is negative it just means speed toward us.

then the formula would be

wavelength ratio = sqrt ((1+V)/(1-V))

$$\sqrt{\frac{1+V}{1-V}}$$

So you make a rainbow with the stars light and look at bright and dark lines corresponding to certain chemical elements glowing when they get hot---the socalled spectrum.
And suppose that the wavelengths of these telltale color patterns are all 1 percent longer than they should be if measured in earth lab.

so then the wavelength ratio must be 1.01
So you have to solve this equation for the speed V

1.1 = sqrt ((1+V)/(1-V))

It is not hard to solve, first you square both sides etc etc.

1.0201 = (1+V)/(1-V)

There is a less accurate formula where you just say
wavelength ratio = 1 + V
So in this case you would say
1.01 = 1 + V and V would be 1 percent of the speed of light.
It is just approximate, the other formula is the correct one.

Now take another case. Suppose the star has no spectral shift----there is no red or blue doppler shift. And suppose you have estimated the distance to the star. Call it R. Some number of lightyears.

Well it isnt getting any closer or farther, but it's position might be changing by some angle measured in radians, some angle theta every year.
So then you can estimate the sideways motion speed.

sideways speed = theta R per year.

If R = 10 lightyears, and theta is 1/1000 of a radian per year. then the sideways speed is
1/100 of a lightyear per year. That is 1/100 of the speed of light.

And then there are combined cases where you have both radial and sideways motion.

And there are more complicated situations where you have clusters of stars, like globular clusters and open clusters, and spiral galaxies. In those cases you may have additional information. So the formulas get more clever and sophisticated. It depends on what you want to know. For example if you look at a spiral galaxy that happens to be seen edge-on
then you know just because stars make roughly circular orbits around the central bulge, you know that on one side they are coming towards, and on the other they are going away. At any given distance out there will be a maximum doppler shift. It is a little more complicated but not too much more. That way you can deduce the orbit speed of stars at that particular distance out.

8. Mar 8, 2009

### Ultrastar 1

Thank you. Now as I said before, such formulas do exist. Just do the research and you will find it.

9. Mar 8, 2009

### Ultrastar 1

I do not see any cites on your article so, don't tell me to cite my responces if you can't cite yours. Also, it could exist because of what the above article posted by marcus says. Please do your research next time. (:

10. Mar 8, 2009

### Nabeshin

Since when do you have to cite highschool physics? Marcus just gave the equations I was talking about, he didn't cite anything either...

11. Mar 8, 2009

### v2kkim

For local motion of stars like a special huge gravitational influence, marcus's posting should be helpful. For far away stars recession due to space expansion, we can relate the recession speed to space scale factor, I mean the redshift reflects how much the space has expanded since its emission from the star, meaning that space expansion causes wave length change. From this way we can get the recession speed but some assumptions may be needed because we do not know exactly about expansion rate change with time.

12. Mar 8, 2009

### Ultrastar 1

Okay, but he has formulas. That is what the original post asked for. Just drop it.