Question

2) (a) Plot, on a log-log scale, the Main Sequence lifetime versus Mass. Use the mass and luminosities for the star types shown on the table below. (b) Also determine a "best-fit straight line" to this data (a best-fit straight line "by eye" is good enough here) with a slope of -2.5 (this was given in the module as a rough, but not bad, approximation to the Mass-Luminosity Relationship. (c) Where does -2.5 fail the worst? (Marks: 6) Note: the relationship crudely "derived" in the module came from a very simplified mass-luminosity relationship. For the objects in the table below both mass and luminosity are given so there is no reason to make this assumption for your plot. You can just scale to the lifetime of the Sun (100 years)

Answer 1

The luminosity of the star is the energy released per unit time. For main-sequence stars, the energy comes from hydrogen fusion and we have:

L = E/t ---------------------(1)

where	L	=	the luminosity of the star
	E	=	energy produced by H burning
	t	=	time

We can use Einstein’s energy-mass equation to calculate the energy produced by hydrogen burning. The mass converted into energy through burning will be a fraction f of the total mass of the star.

E = f M c² where   ---------------------(2)

where	E	=	energy produced by H burning
	f	=	fraction of mass converted into energy
	M	=	mass of the star
	c	=	speed of light

Combining the last two equations, we have the following expression for the main sequence lifetime:

t_MS ∼ M/L -------------------------------------------------------------------- (3)

Using the mass-luminosity relationship for main sequence stars:

L ∼ M^3.5 -------------------------------------------------------(4)

and substituting for L, we have the expression for main sequence lifetime in terms of stellar mass:

t_MS ∼ M^-2.5   ---------------------(5)

This can be expressed (as above) in solar units:

  ---------------------(6)

tus ~()-2.

where	t⊙	=	Sun MS lifetime = 1010
	M	=	mass of star
	M⊙	=	solar mass

Note: this expression is an approximation only, and not valid for very massive or very light stars. The main limitation is the use of the single value mass-luminosity relationship for main sequence stars.

We will use equation 3 to derive main sequence lifetime as in column I of spreadsheet attached in photo.

File Edit View Insert Format Tools Window Help Chart Area 9 l l. chart wall 3 d . 2 E . I lol All Axes B F G H J K L M N O P Q R S Mass Mv U-B B-V s1 A 1 spectral type Teff 2 05 08 4 BO 5 B5 Mass vs main sequence lifetime 23 17.5 8.4 5.9 7 FO | 8 GO | 9 KO 10 K5 11 MO T 12 M5 13 M8 C Luminosity R 44500 790000 35800 170000 30000 52000 15400 830 9520 54 7200 6.5 6030 1.5 5250 0.42 4350 0.15 3850 0.077 3240 0.011 2640 0.0012 4.1 2.7 1.6 1.1 0.79 0.68 0.63 0.33 0.17 2.9 1.6 1.05 0.79 0.67 0.51 0.21 0.06 -5.7 -4.9 -4 -1.2 0.6 2.7 4.4 5.9 7.4 8.8 12.3 16 -1.19 -1.14 -1.08 -0.58 -0.02 0.03 0.06 0.45 0.98 1.22 1.24 1.53 I t=(M/L) -0.33 7.59494E-05 -0.32 0.000135294 -0.3 0.000336538 -0.17 0.007108434 -0.02 0.053703704 0.3 0.246153846 0.58 0.7 0.81 1.880952381 1.15 4.466666667 1.4 6.623376623 1.64 19.09090909 1.93L Lifetime títsun) t=(M/L) .. Approximation for slope = -2.5 .. 0.001 0.0001 1E-05 0.01 0.1 100 Mass M(Msun)