How Does Background Star Distance Affect Parallax Measurements?

[ltjrpliskin]

Homework Statement

"Parallaxes are measured relative to background stars. If these are not infinitely distant
themselves, then the parallax to the foreground object will be underestimated
and its distance will be overestimated.
Calculate the distance that will be measured to a star at a true distance of 40 pc if
the background stars are at a distance of 400 pc and this effect is not allowed for."

I looked through my book and even the lecture slides. It doesn't explain what true distance is...
or am I missing something really key here?

 

[cepheid]

Homework Statement

"Parallaxes are measured relative to background stars. If these are not infinitely distant
themselves, then the parallax to the foreground object will be underestimated
and its distance will be overestimated.
Calculate the distance that will be measured to a star at a true distance of 40 pc if
the background stars are at a distance of 400 pc and this effect is not allowed for."

I looked through my book and even the lecture slides. It doesn't explain what true distance is...
or am I missing something really key here?

Not sure what you are asking here. The true distance is the true distance -- i.e. how far away the object actually is.

The point of the question is that the distance that you measure (using parallax) may not actually be the true (correct) distance. In other words, your measurement is wrong -- it has some error, because you assumed that the background objects were fixed. You didn't take into account that the background objects would also shift around due to parallax (just less perceptibly).

 

[ltjrpliskin]

Not sure what you are asking here. The true distance is the true distance -- i.e. how far away the object actually is.

The point of the question is that the distance that you measure (using parallax) may not actually be the true (correct) distance. In other words, your measurement is wrong -- it has some error, because you assumed that the background objects were fixed. You didn't take into account that the background objects would also shift around due to parallax (just less perceptibly).

I see, that makes more sense. I was thinking silly stuff.
But one thing I don't understand is how I can measure the parallax distance with just the information about the background stars being at a distance of 400pc.

 

[cepheid]

I see, that makes more sense. I was thinking silly stuff.
But one thing I don't understand is how I can measure the parallax distance with just the information about the background stars being at a distance of 400pc.

What would be the parallax angle of the 40 pc object if this shift were measured relative to to a truly fixed background object?

What would be the parallax angle of the 400 pc object if this shift were measured relative to to a truly fixed background object?

So, what is the angle between the 40 pc object and the 400 pc object (which you're taking to be the 40 pc object's parallax angle), and how much smaller is this than the actual parallax angle for the 40 pc object?

 

[ltjrpliskin]

What would be the parallax angle of the 40 pc object if this shift were measured relative to to a truly fixed background object?

What would be the parallax angle of the 400 pc object if this shift were measured relative to to a truly fixed background object?

So, what is the angle between the 40 pc object and the 400 pc object (which you're taking to be the 40 pc object's parallax angle), and how much smaller is this than the actual parallax angle for the 40 pc object?

Thanks I think I understand it now! Since the background stars are not infinitely distant "the foreground parallax is underestimated" so the parallax angle is actually smaller (in this case 1/40 - 1/400) which gives us the overestimated distance of 400/9 pc.

 

[cepheid]

Sounds about right to me