1) From the ground of the Earth, atmospheric turbulence limits the typical best image to a full-wide-at-half-maximum (FWHM) of ~0.5 arcsceond (this is an angle). Suppose we can determine positional accuracy of stars to ~1/10 of the FWHM, to how far in light years can we use the parallax method to measure the distance to stars?

Now I am stuck on this question. My biggest trouble is that I don't know how to relate the 0.5 arcsceond to the parallax method or parallax angle. Can someone please help me out?

Any help is greatly appreciated!

Big test tomorrow...Can someone please help me?

Since you have a test tomorrow and no one has answered, I'll give it a shot. IIRC, the distance in parsecs is inversely related to the parallax angle in arc seconds. So parallax of say .5 arc secs equalls a distance of 2 parsecs. I don't recall off hand how long a parsec is, serveral ly I believe.

Now for your problem. I don't care for the way it's worded. A little ambiguous to my thinking. Anyway, it's saying that the maximum accuracy is 1/10 of .5 arc sec, or .05 arc sec. This would occur for a distance of 20 parsecs. That distance expressed in ly is the answer your text is asking for unless I am mistaken.

edit>> 1 parsec = 3.262 ly according to Wikipedia.

For the parallax method, we need two angles and the length of the baseline...

But there is only one angle (0.5 arcsceond) given in this problem...how can we find distance using just one angle? I am very confused...

By the way, thanks for your reply~!

I don't get the setup of the problem!

For example, in a diagram like this, WHERE exactly is the angle being referred as 0.5 arcsecond?
http://www.geocities.com/paulntobin/image/parallax.gif

The angle ANB is the parallax angle. Since the baseline is always the diameter of the Earth's orbit, that's all you need. Now for a 1 arc sec parallax, the distance to N is 1 parsec (that's the definition of the parsec) which equals 3.262 ly. Now keeping in mind that there is a reciprical relationship between the angle and the distance to star N (in parsecs) then .05 arc secs (.5 * 1/10) equals 20 parsecs.

Basically what's going on is that we're using some small angle approximations to simplify things. The line segments AN and BN are taken to be equal, as are angles NAB and NBA. That way the geometry is describable with only the parallax angle. This only works for distant objects, where the diameter of the Earth's orbit is much less than the distance to the object.

Here's the Wiki for parsec, it may help explain it better than I.
http://en.wikipedia.org/wiki/Parsec

Edit>> Technically, the parallax angle is 1/2 of angle ANB. And the baseline is half of the diameter of the Earth's orbit. That shouldn't change the answer, as that is assumed to be the way the original .5 arc sec was measured.

http://burro.case.edu/Academics/Astr221/Light/parallax.gif

The parallax angle is labelled "p" in the above figure.

When we talk about distance to galaxies or stars, is it the distance measured from the Sun or is it the distance from the Earth?


"that angle can be shown to be equal to the angle you have in your main diagram labelled "Nearby star "N" " <----can you please explain why there are equal?

Isn't the answer to question 1: d=(1/0.025) pc ? (becaue the 0.5arcseconds is the subtended angle of the apparent positions of the star and the parallax angle is 1/2 of it...?)

The distance to even the nearest star, Alpha Centari, is very large compared to the dimensions of the Earth's orbit, so in practice no distinction is made as to whether it refers to the distance from the Earth or Sun. The difference is hidden by the practical limits of measurement. But that is a valid point. If you were trying measure the distance to something much closer, it would matter. In practice I'm sure a lot of corrections are made. For example, if the direction to the star is very far from right angles to the baseline AB. The problem is describing a very idealized situation.

I don't think they're equal. p should be 1/2 the angle measured (comparing at point A and B. In other words p= 1/2 angle ANB.) Originally I had forgotten that the baseline for the standard parsec is 1/2 the Earth's orbital diameter. Sorry if that caused any confusion.

I think you're right, d=1/.025pc.

Oh, I see! Thanks!

Also, I have another question. How come we are unable to use the parallax method to measure the distance to stars when the full-wide-at-half-maximum is less than 0.5 x 1/10 = 0.05 arcseconds? I just don't get why...

Thanks for helping me!

Well, as defined by the problem that is the limit of error in the measurement, so basically the distance to star N is somewhere between the distance corresponding to .05 arc sec and infinity, not exactly a precise distance. ;)