Astronomy 12 - Spring 1999 (S.T. Myers)

1. We use the usual relation

   log( L / L_sun ) = ( 4.83 - M_v ) / 2.5

   so

   4.83 - M_v 2.5 [ 1.15 log( P / 1 day ) + 2.47 ]

   or

   M_v = -2.875 log( P / 1 day ) - 1.34

   for the absolute visual magnitude of a Cepheid. The distance modulus

   m_v - M_v = 5 log( d / 1 pc ) - 5 + A_v       ( no dust A_v=0 )

   gives

   log( d / 1 pc ) = 0.2 [ m_v + 2.875 log( P / 1 day ) + 6.34 ] = 0.2 m_v + 0.575 log( P / 1 day ) + 1.27

   thus for Polaris ( P = 13 days, m_v = 2.3 )

   log( d / 1 pc ) = 0.2 · 2.3 + 0.575 · log 13 + 1.27 = 2.37

   or d = 235 pc. Note that the absolute magnitude of Polaris is

   M_v = -2.875 log 13 - 1.34 = -4.54

   with a luminosity of 5611 Lsun!

2. The mean density

   = ( 3 M ) / ( 4 R³ )

   and the period-density relation

   P = [ 3/ (2G) ]^1/2

   give on substitution

   P = [ ( 2 ² R³ ) / ( G M ) ]^1/2

   or numerically

   P = 6117 s · ( M / M_sun )^-1/2 ( R / R_sun )^3/2 ( 3 / 4 )^-1/2

   which fills in our table:

   M / Msun	R / Rsun	Mv	P (days)
   5	36.3	-3.1	6.925
   10	195.0	-5.9	60.96

   The quantities of interest are:

   log P	Mv
   0.840	-3.1
   1.785	-5.1

   and thus we solve the system of equations

   -3.1 = 0.840 · a   +   b
   -5.9 = 1.785 · a   +   b

   which gives by subtraction

   2.8 = - 0.945 · a   ==>   a = -2.963

   and then

   b = -3.1 - 0.840 · a   ==>   b = -0.611

   by substitution. Our period-magnitude relation is thus

   M_v = -2.963 log( P / 1 day ) - 0.61

   (which is similar to that used in the previous problem), and

   P = 20 days   ==>   log P = 1.301   ==>   M_v = -4.465

   and thus the distance modulus

   m_v - M_v = 10.4 - (-4.465) = 14.865

   gives us

   log( d / 1 pc ) = 0.2 [ m_v - M_v + 5 ] = 3.973   ==>   d = 9397 pc

   and the star is almost 10 kpc away.

3. To locate Mira on the H-R diagram, we need its luminosity (in Lsun) and its effective temperature. For an absolute magnitude M_bol = -5 and the Sun's bolometric magnitude M_bol = +4.75 we get

   log( L / L_sun ) = ( 4.75 + 5 )/2.5 = 3.9   ==>   L = 7943 L_sun

   Looking at the H-R diagram on p.321 of ZG4 for example, the point

   log( L / L_sun ) = 3.9         log( T ) = 3.36

   is to the right of the asymptotic giant branch (AGB). Mira must be an evolved star (see also the diagram on p.353 of ZG4).

   For the average temperature T = 2300 K, the average radius is

   R / R_sun = ( L / L_sun)^1/2 ( T / 5770 K )^-2 = 439

   which is in line with it being a late M supergiant. The luminosity ratio is given by the magnitude difference

   L_max / L_min = 10 ^5.1/2.5 = 109.6

   and so

   R_max / R_min = ( L_max / L_min )^1/2 ( T_max / T_min )^-2 = 6.2

   and the radius changes by a factor of 6 during the pulsation cycle!

4. For stars of the same luminosity L, they will appear to have a flux

   f = L / (4d²)

   and thus counting the number brighter than some limiting flux f is equivalent to counting the number closer than a limiting distance

   d = [ L / (4f) ]^1/2.

   If the stars are distributed uniformly with density n per unit volume, then

   N( >f ) = n V( d³ ( f^-1/2 )³ f^-3/2.

   Note that this is true only for a uniform volume density of sources, but is true for any luminosity function of stars.

5. The circular velocity is given by

   v_circ² = G M / R   ==>   M v² R

   and since the velocity v is constant

   M( < R ) R

   is the mass law. For a spherical mass "halo", the mass grows as

   dM/dr = 4R²   ==>   R^-2 dM/dr R^-2

   which is a common profile for the outer parts of massive galaxies and clusters of galaxies. Note you could have also obtained this scaling by

   M / R³ R / R³ R^-2

   as before.

smyers@nrao.edu   Steven T. Myers