On Friday we studied a method of measuring large distances that cannot be measured directly. This involved a process of observing two distant objects from two locations a measurable distance apart. When these two observations are made, the near object appears to jump in relation to the far object. For your write-up on this section describe the example that we covered in class. What measurements did we make? How close were our answers to the actual distances? What were the possible sources of error? Can you imagine a more effective way to measure angular separation? Draw a picture of the relationship.
We will continue this discussion on either Monday or Tuesday as we incorporate the small angle approximation and parsecs.
Post questions or thoughts in the comments below.
Mr. H
We continued this by adding in the small angle approximation formula. This tells us that for angles smaller than 0.1 degree the tan (x) = x (as long as x is in radians and not degrees), therefore we do not need to use the tangent function anymore. So we can go from using this formula:
to this one:
, using the notation from
class where ½ (a+b) = half of the total parallax shift. From this point forward, let’s call it angle α. NOTE:
for this formula, the above angle must be expressed in radians. The formula for converting degrees to radians
is: 
. Solving this equation for rad (radians) we
get: 
. If I substitute this into the above equation,
and then solve for d (distance), we get:
. If we know angle α in radians (let’s call it αrad),
we can just use the simple formula: 
. We then went through the derivation of a new
unit called the parsec (this stands for parallax of 1 second of arc (arcsec)). We figured out that if 1 arcsec was 1/3600 of
1 degree that it has a corresponding distance of 31,000,000,000,000 km. This unit is a parsec. 1 parsec=31 trillion km=3.26 light years (ly).
But, if 1 second of arc = a distance of 1
parsec, then with this new unit we can simplify things greatly. If we plug our angle in arcseconds to the
formula above we get our answer with math that we can do in our heads.
Examples:
for 1 arcsec: 
therefore d=1 parsec. For 2 arcsec:
. For 0.5 arcsecs: 
Hopefully this makes sense; the larger the
parallax measurement (the apparent shift of the stars), the nearer the
distance. Parsecs makes this measurement
easy and convenient.
therefore d=1 parsec. For 2 arcsec:
. For 0.5 arcsecs: Hopefully this makes sense; the larger the
parallax measurement (the apparent shift of the stars), the nearer the
distance. Parsecs makes this measurement
easy and convenient.
I just realized that the formulas are not posting here. To see these, go here.
