We use the usual relation

log( L / L _{sun}) = ( 4.83 - M_{v}) / 2.5so

4.83 - M _{v}2.5 [ 1.15 log( P / 1 day ) + 2.47 ]or

M _{v}= -2.875 log( P / 1 day ) - 1.34for 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 ) + 1.27thus 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 = -4.54with a luminosity of 5611 Lsun!

The mean density

= ( 3 M ) / ( 4 R ^{3})and the period-density relation

P _{ }= [ 3/ (2G) ]^{1/2}give on substitution

P = [ ( 2 ^{2}R^{3}) / ( 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 / M _{sun}^{ }R / R _{sun}^{ }M _{v}^{ }P (days) _{ }^{ }5 ^{ }_{ }36.3 ^{ }_{ }-3.1 ^{ }_{ }6.925 ^{ }_{ }10 ^{ }_{ }195.0 ^{ }_{ }-5.9 ^{ }_{ }60.96 ^{ }_{ }The quantities of interest are:

log P _{ }M _{v}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 + bwhich 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), andP = 20 days ==> log P = 1.301 ==> M _{v}= -4.465and thus the distance modulus

m _{v}- M_{v}= 10.4 - (-4.465) = 14.865gives us

log( d / 1 pc ) = 0.2 [ m _{v}- M_{v}+ 5 ] = 3.973 ==> d = 9397 pcand the star is almost 10 kpc away.

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 getlog( 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 ) = 3.36is 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}= 439which 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}= 109.6and so

R _{max}/ R_{min}= ( L_{max}/ L_{min})^{1/2}( T_{max}/ T_{min})^{-2}= 6.2and the radius changes by a factor of 6 during the pulsation cycle!

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

f = L / (4d ^{2})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, thenN( >f ) = *n*V(d ^{3}( 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.

The circular velocity is given by

v _{circ}^{2}= G M / R ==> M v^{2}Rand since the velocity v is constant

M( < R ) R is the mass law. For a spherical mass "halo", the mass grows as

dM/dr = 4R ^{2}==> 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 / R ^{3}R / R^{3}R^{-2}as before.

*smyers@nrao.edu*
*Steven T. Myers*