1. Way back in 1838, Bessel determined the parallax to 61 Cygni to be
    0.314 arcseconds. How good was his measurement? You can check it against a
    more modern measurement by the Hipparcos satellite.  

    a) The star 61 Cygni
    appears in the Hipparcos catalog as star number 104214. Use the Hipparcos
    catalog search site to find the star's parallax, as measured by Hipparcos;
    type this number into the "Hipparcos identifier" box down near the bottom of
    the form.  

           parallax angle = 287 milliarcseconds  = 0.287 arcsec


    b) What is the percentage difference between the Hipparcos angle
    and Bessel's angle?  


                            0.314 arcsec - 0.287 arcsec 
           percent diff  =  ---------------------------  * (100 percent)
                                  0.287 arcsec


                         =  9.4 percent


    c) Use the Hipparcos parallax angle to calculate the
    distance to the star.  

                             1
      Hip distance  =  ------------  =  3.48 pc
                       0.287 arcsec


    d) What is the percentage difference between the
    Hipparcos distance and Bessel's distance? 

                               1
      Bessel distance  =  ------------  =  3.18 pc
                          0.314 arcsec


                          (3.18 pc - 3.48 pc)
      percent diff     =  -------------------  * (100 percent)
                             3.48 pc


                       =   -8.6 percent 

      Note that this is not exactly the same as the percentage
      difference in parallax angle.  The reason is that when
      we take the reciprocal of the angle, we do funny things
      to the uncertainties.



   2. Computing the uncertainty in distance to a star based on its parallax
      can be a tricky thing. Consider the case of Joe and the star Rigel. Joe
      measures a parallax half-angle of

                pi = 0.004 arcsec 

      to Rigel. He estimates the
      uncertainty in his measurement to be +/- 0.003 arcsec. 
      In other words, the parallax could be as large as 0.007 arcsec, 
      or as small as 0.001 arcsec.

      a. What is the distance to Rigel, if Joe's measurement is exactly
         correct?

                                 1
                distance  =  ------------   =   250 pc
                             0.004 arcsec


      b. What is the distance to Rigel, if the error in Joe's measurement
         is +0.003 arcsec (i.e. if Joe's measurement is at the lower 
         end of its estimated uncertainty)?

         
             If the true angle is 0.004 - 0.003 = 0.001 arcsec, then

                                 1
                distance  =  ------------   =   1000 pc
                             0.001 arcsec


      c. What is the distance to Rigel, if the error in Joe's measurement
         is -0.003 arcsec (i.e. if Joe's measurement is at the upper end of its
         estimated uncertainty)? 

             If the true angle is 0.004 + 0.003 = 0.007 arcsec, then

                                 1
                distance  =  ------------   =    143 pc
                             0.007 arcsec


             Notice that this means that a nice, symmetric uncertainty
             in the angle

                                    smallest    best     largest

               angle could be       0.001       0.004      0.007    arcsec


             leads to quite an asymmetric uncertainty in the position

                                    smallest    best     largest

             distance could be       143         250      1000      pc    





      Bonus! What is the actual distance to Rigel? How well do we know it? 

         SIMBAD says that the parallax to this star is

                  parallax  =     4.22  +/-     0.81  milliarcsec

                            =  0.00422  +/-  0.00081  arcsec

         which leads to distances of


             min  distance  =    1 /  (0.00422 + 0.00081)  =   199 pc

            best  distance  =    1 /  (0.00422)            =   237 pc

             max  distance  =    1 /  (0.00422 - 0.00081)  =   293 pc


         So the distance lies somewhere between about 200 and 300 pc.