ASTR1040 Spring, 2005
Due date: Thursday, March 10
1.a. Estimate the hydrogen-burning lifetime of a star of 15 solar masses and a luminosity 10,000 times greater than the solar luminosity.
b. What would your estimate be for the lifetime of a star having a mass of 0.2 solar masses and a luminosity of 0.008 times the solar luminosity?
c. If 10 percent of the star's original mass is now in the form of helium in the core, calculate the helium-burning lifetime of the star. To do this, you must find the fraction of the helium mass that is converted into energy by E = m^c2, which you can do by comparing the mass of a carbon nucleus with the sum of the masses of three helium nuclei (the helium mass is 4.002603 u, and the carbon mass is 12.000000 u). Then you can calculate the total amount of energy available to the star through helium-burning and divide this by the star's luminosity to get the lifetime (in seconds). Assume that the luminosity of the star during its helium-burning phase is 100 times its main-sequence luminosity (Note: The u stands for the basic atomic mass unit, whose value is 1.6606 X 10^-27 kg - BUT: You don't need this value in order to work this problem! (Think in terms of ratios...)
2. Suppose that when the Sun becomes a red giant it will increase its radius by a factor of 100 while its surface temperature drops to half of its current value.
a. What will be the Sun's luminosity at that point, compared to its current luminosity (i.e., expressed as a ratio) and in basic units (i.e., watts)?
b. What will the surface gravity be (expressed as a ratio with the Sun's current surface gravity)?
c. What will the escape speed be from the surface of the Sun as a red giant? Compare with the escape speed now.
d. If you have a telescope capable of detecting stars as faint as apparent visual magnitude m_v = +20, how far away could you see this star? (Hints: You need to convert the luminosity of the star, which you calculated in part a, to an absolute visual magnitude, and then use the distance modulus relation with m_v = +20 to find the distance. To calculate the absolute visual magnitude from the absolute bolometric magnitude, you need the bolometric correction, which in this case you should assume to be B.C. = -2.0; i.e., the bolometric magnitude is two magnitudes brighter than the visual magnitude.)
3. Now suppose that the Sun loses its outer envelope due to mass loss during its red giant phase, leaving behind a core (a white dwarf) having a mass equal to half the original main sequence mass, a radius equal to 0.01 of the original radius, and a surface temperature equal to twice the original temperature of 5780 K.
a. What are the luminosity and bolometric absolute magnitude of the white dwarf remnant?
b. If the star's bolometric magnitude is -0.4 magnitudes brighter than its visual magnitude (i.e., its bolometric correction is BC = -0.4), calculate how far away this remnant can be detected with a telescope whose limiting visual magnitude is +20. Compare your answer with the distance you found in 2.d., above.
c. Calculate the average density (converted to units of grams/cm^3) and the escape speed for the white dwarf.
d. Calculate the wavelength of maximum emission for the white dwarf. What kind of telescope would be best for observing it?
4. A star of initial mass 30 solar masses loses mass at a rate of 5 X 10^-6 solar masses per year during its main sequence lifetime of 3 X 10^6 years. Then it blows up in a supernova explosion, leaving behind a remnant whose mass is 1.6 solar masses. The absolute visual magnitude of the supernova at its peak is M_v = -19.0.
a. What is the star's mass just before it blows up?
b. If the mass blown off in the explosion travels outward with an average speed of 2000 km/sec, what is the kinetic energy of the explosion?
c. If the bolometric correction at peak brightness is B.C. = -1.0 (i.e., the bolometric absolute magnitude is 1.0 magnitudes brighter than the visual absolute magnitude), and the duration of the peak is 3 days, how much energy is radiated away during this time? Compare this with the kinetic energy you found in part a.
d. Assume that the energy released in the form of neutrinos is 100 times greater than the sum of the kinetic plus luminous energy of the explosion. How does the total energy (including all three forms) released in the supernova explosion compare with the total energy the Sun can produce over its entire hydrogen-burning lifetime?
e. If the remnant has a radius of 10 km, calculate its average density (expressed in units of grams/cm^3). Compare your value with those for the white dwarf and for the Sun before it became a red giant.
f. If the remnant's surface temperature is 10^6 K, what is its luminosity? If its bolometric correction is B.C. = -3.0 (i.e., the bolometric magnitude is 3.0 magnitudes brighter than the visual magnitude), how far away can this remnant be detected by our telescope whose limiting visual apparent magnitude is m_v = +20?
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