Astronomy 11 - Fall 1998 (S.T. Myers)

1. For 0.8 Msun and 0.87 Rsun, the escape velocity is

   v_esc = ( 2GM/R )^1/2 = 594 km/s

   which implies

   K/m = 0.5 v_esc² = 1.76 x 10¹¹ J/kg

   for the kinetic energy per unit mass deposited. Thus the required impact rate is

   L / (K/m) = (1.532x10²⁶ J/s)/(1.76x10¹¹ J/kg) = 8.69 x 10¹⁴ kg/s

   which is equivalent to 1.37 x 10^-8 Msun/yr, and thus the total mass of the star 0.8 Msun would be deposited in 58 Myr. This timescale is way too short compared to the billion-year lifetimes of stars.

2. You can extract energy from a sphere of mass M and radius R by contracting it, and thus making its potential energy more negative. This energy can be released as heat (and thus blackbody luminosity) at the rate

   L = -dU/dt = -d/dt[ -GM²/R ] = - GM²/R² dR/dt

   where we have used the chain rule dU/dt = (dU/dR)(dR/dt). Thus dR/dt is negative (contraction) and equal to

   -dR/dt = L / ( GM²/R² ) = RL/U

   where the binding energy

   U = GM²/R = 2.82 x 10⁴¹ J

   implies

   L / U = 5.43 x 10^-16 s^-1

   and thus the contraction rate is 5.43 x 10^-16 of the radius per sec, or 10.4 m per year!

   The timescale is given by

   U / L = 1.84 x 10¹⁵ s = 58.3 Myr

   which is what we got in #1. Why? Because the kinetic energy release in accretion from infinity is by definition due to the gravitational potential! Still, its too short...

3. The energy per unit mass in the mass itself is

   E/m = c² = 9 x 10¹⁶ J/kg = 1.506 x 10^-10 J/amu.

   The mass defect is

   4 ¹H - ⁴He = 4.03130 - 4.00268 = 0.02862 amu

   and

   0.02862/4.03130 = 0.0071    ==>    0.0071 c² = 6.39 x 10¹⁴ J/kg

   is released during the fusion. Each fusion generates

   1.506 x 10^-10 J/amu x 0.02862 amu/fusion = 4.31 x 10^-12 J/fusion

   and thus the fusion rate must be 3.55 x 10³⁷ fusions/sec for a luminosity of 0.4Lsun. For a stellar mass of 0.8Msun, there is 0.6Msun of hydrogen and thus 0.06Msun hydrogen in the core, or 1.2 x 10^29 kg. This must be fused at the rate 0.4Lsun

   L / 0.0071c² = (1.53 x 10²⁶ J/s)/(6.39 x 10¹⁴ J/kg) = 2.40 x 10¹¹ kg/s

   which for 1.2x10^29 kg of core hydrogen will be able to go on for 15.9 Gyr! Note that if you do exactly the same calculation for 1Lsun and 1Msun you get 8 Gyr, and thus smaller stars burn their fuel slower and thus live longer.

4. The mass defect to carbon is

   12 ¹H - ¹²C = 12.0939 - 12 = 0.0939 amu

   or

   0.0939/12.0939 = 0.00776 c² = 6.99 x 10¹⁴ J/kg

   is the total energy yield per unit mass. Therefore, 1.4Msun of hydrogen will release 1.96 x 10^45 J of energy (ignoring the fact that only 75% of that mass was hydrogen to begin with, and you don't get as much energy out fusing helium to carbon). The gravitational potential energy for R=2000 km is

   U ~ GM²/R = 2.61 x 10⁴⁴ J    ==>    E / U ~ 7.5

   and thus even though a white dwarf has a higher binding energy per unit mass (U/M ~ GM/R) than normal stars, nuclear energy generation still dominates.

5. We calculated a fusion rate of 3.55 x 10^37 fusions per second for EPS451 (0.4Lsun) and thus our Sun (1Lsun) must fuse at the rate of 8.88 x 10^37 per second, and with each fusion producing two neutrinos the rate of neutrino production must be 1.8 x 10^38 per second! Now at a distance of r=1 AU (1.5 x 10^11 m) the flux is

   (1.8 x 10³² s^-1)/ (4r²) = 6.3 x 10¹⁴ ( r / 1 AU )^-2 m^-2s^-1

   and thus around 600 trillion neutrinos from the core of the Sun pass through our bodies each second!

smyers@nrao.edu   Steven T. Myers