Barnard's Star is a low-mass dwarf on the main sequence which happens to be
  relatively close to our solar system. Assume that its mass is about 0.16
  solar masses.

     1. what is the apparent magnitude of Barnard's Star?

            apparent V mag = 9.54


     2. what is the distance to Barnard's Star?

            Hipparcos finds parallax = 549.3 milliarcseconds

            -> distance  =  1/0.5493  =  1.82 pc


     3. what is the absolute magnitude of Barnard's Star?

            (m - M)   =  5 log d  -  5

                  M   =  m     +  5  -  5 log d
 
                      =  9.54  +  5  -  5 log(1.82)

                      =    14.54     -  1.30

                      =    13.24


     4. what is the luminosity of Barnard's Star?

             If a star has absolute V-band magnitude M,
             then its luminosity is roughly

                               28            -0.4 * M
                L     =  3 x 10   W     *  10

                               28                   -6
                      =  3 x 10   W     *  5.06 x 10

                      
                      =  1.5 x 10^(23) W


             The Sun produces about 3.8 x 10^(26) W, so
             Barnard's star has a luminosity of only
             about 0.0004 solar.


     5. if Barnard's star were pure hydrogen at birth, 
        and maintained its current luminosity, roughly how long 
        could it remain on the main sequence? Use its initial mass, 
        its luminosity, and the efficiency of hydrogen fusion to make 
        your estimate. 

             Barnard's star produces 1.5 x 10^(23) Watts,
             or 1.5 x 10^(23) Joules each second.

             We know that fusion of 4 hydrogen atoms
             requires 6.69 x 10^(-27) kg and generates
             4.3 x 10^(-12) Joules.  So we can figure out
             how many kg of hydrogen Barnard's star must
             fuse each second.


                  6.69 x 10^(-27) kg           4.3 x 10^(-12) J
                 --------------------   =    --------------------
                       X kg                    1.5  x 10^(23) J


             Solve for X, the number of kg which must be fused
             each second:

                       X kg   =  2.36 x 10^(8) kg

             Now, if Barnard's star has a mass of 0.16 solar,
             and if it was all pure hydrogen at birth, then it
             started with

                      mass    =  0.16  *  2 x 10^(30) kg

                      mass    =  3.2 x 10^(29) kg
           
             We can calculate how long it would last if it 
             fused X kg each second:

                                     starting mass
                lifetime      =   -------------------
                                  rate of using mass

                                    3.2 x 10^(29) kg
                              =   -------------------
                                    2.36 x 10^(8) kg/s


                              =    1.4 x 10^(21)  seconds

                              =    43,000 billion years



 2.  Look at the figure at the link below, showing
     the expanding shell around SN 1993J.
     Calculate the speed with which gas was ejected
     in the supernova explosion.

          My trusty ruler shows that

              1993 May 17     radius =    220 AU     JD = 2,449,125

              1997 Jun 7      radius = 11,000 AU     JD = 2,450,607
              ------------------------------------------------------
                                       10,880 AU              1,482


           So the shell was expanding at a rate of

                             10,880 AU       8.5 x 10^(-5) AU
               avg speed  =  ----------   =  ----------------
                              1,482 days           second


                          =  13,000 km/sec