1) by definition, a main sequence star is one that fuses hydrogen
(H) to helium (He) in the core
(the inner 15% by mass) of the star
2) the lifetime of a main sequence star is determined by
Tms = (amount of energy available)/(rate
of energy use)
Tms = Mstar
fcore fH fmass lost c2/Lstar
where Mstar = the mass of the star
fcore = the fraction of the star's mass that
is in the core, i.e., that is converted
from H to He during the main sequence phase = 0.15
fH = the original fraction of the star that is hydrogen fuel = 0.75
fmass lost = fraction of the mass lost in a H --> He fusion reaction = 0.007
Lstar = luminosity of the star = rate at which energy is radiated
3) for main sequence stars, the mass and the luminosity are related:
Lstar = Mstar4
as long as both Lstar and Mstar are in solar units
4) combining the two previous equations, and substituting numerical values for the constants
Tms = 1010 yrs/(Mstar)3
5) note that the larger the mass, the less time a star spends on the
main sequence; therefore, in a cluster
(for which all stars were born at the same time, and therefore have
the same age), the hotter and brighter stars on the main sequence leave
first for the red giant region (where they will fuse H to He in their outer
envelope and also fuse He to C in their core)
furthermore, the age of the cluster can easily be found, because it is also the lifetime of the star just now leaving the main sequence... so to find the age of a cluster, use your H-R diagram to find the B-V color index of the star just leaving the main sequence; then use the stellar properties table to convert this to the mass of the star just leaving the main sequence
6) finally, use the relationship in #4 above to determine the main sequence
lifetime of that star and, therefore, the age of the cluster