Heliocentric parallax

The simplest way to measure the distance to an object via parallax is to make simultaneous measurements from two locations on Earth. However, as we saw last time, this method only works for objects which are relatively nearby:

       Object        maximum possible parallax
                         (arcseconds)
   -------------------------------------------------
       Mars                 34
       Jupiter               4
       Neptune               0.6
       Proxima Centauri      0.00007
   -------------------------------------------------

In short, simultaneous two-location measurements will only work for bodies within our own solar system. We need to find some larger baseline to measure the parallax to other stars....

A longer baseline: the orbit of the Earth around the Sun

Astronomers need a VERY long baseline in order to produce a parallax angle which is large enough to detect via conventional imaging. Fortunately, there is a convenient baseline just waiting to be used -- if we are willing to discard the requirement that measurements be simultaneous (more on this later). This longer baseline is the radius of the Earth's orbit around the Sun:

Note the convention used only in this situation: the quoted "heliocentric parallax angle" π (pi) is always HALF the apparent angular shift.

So, if we measure a parallax half-angle π (pi) to a star, we can calculate its distance very simply:

                     R
         L   =   ---------
                  tan(π)
where
                           6
         R   =   149.6 x 10   km

             =   1 Astronomical Unit (AU)


Small angles and peculiar units make life easy

Now, for small angles -- and the angles are always small for stars -- we can use the small angle approximations:

         tan(π) ~=  sin(π)  ~=  π
as long as we measure the angle in radians. Even if we don't use radians, it still remains true that cutting a small angle in half will also cut its tangent in half
               π           1       
         tan( --- )  ~=   --- tan( π )
               2           2
and, in general, there is a linear relationship between the tiny angle π and its tangent, regardless of the units. Take a look for yourself:
      angle        angle        tan(angle)    
    (degrees)    (radians)       
   --------------------------------------------
       1          0.017453        0.017455
       0.5        0.008727        0.008727
       0.1        0.0017453       0.0017453
       0.01       0.00017453      0.00017453
   --------------------------------------------

Okay, fine. Why am I belaboring this point? Because astronomers have chosen a set of units for parallax calculations which look strange, but turn out to simplify the actual work. The relationships between these units depend on the fact that there is a simple linear relationship between a tiny parallax angle, in ANY units, and the tangent of that angle.

angles are measured in arcseconds (")
Recall that 1 arcsecond = 1/3600 degree. Astronomers use arcseconds because parallax values for nearby stars are just a bit less than one arcsecond. You may also see angles quoted in mas, which stand for milli-arcseconds: 1 mas = 1/1000 arcsecond.

distances are measured in parsecs (pc)
A parsec is defined as the distance at which a star will have a heliocentric parallax half-angle of 1 arcsecond.


  Q: How many meters are in one parsec?  
     How many light years are in one parsec?

Given these units, and the linear relationship between a small angle and its tangent, we can calculate the distance to a star (in pc) very simply if we know its parallax half-angle in arcseconds:

                                 1
             distance (pc)  =  -----
                                 π

Give it a try: the first star to have its parallax measured accurately was 61 Cygni. Way back in 1838, the German astronomer Friedrich Bessel announced that its parallax half-angle was 0.314 arcseconds.


  Q: Based on Bessel's measurement, 
     what is the distance to 61 Cygni?


Some observational difficulties

Actually, 1838 isn't all that long ago. Astronomers had been looking through telescopes since the time of Galileo, in the early 1600s. Why did it take over 200 years for someone to measure the parallax to another star?

There are several reasons:


The bottom line

The best large set of parallax measurements come from the Hipparcos satellite, which measured the position and brightness of relatively bright (brighter than tenth magnitude or so) stars over the entire sky during the period 1989-1993. A large team of scientists turned its millions of raw measurements into a consistent catalog of distances and luminosities.

The precision of the Hipparcos measurements of parallax is about 0.001 arcsecond. You might think that such precision would allow us to measure distances to stars as far away as 1000 pc. However, as we shall see, it turns out that the true range for accurate distances is quite a bit smaller.