The number of He nuclei produced each second corresponds to the number of fusion reactions occuring (as each reaction produces one He nucleus). Divide the energy output of the Sun by the energy of one fusion reaction to find out how many reactions per second are required to power it, and we'll have our Helium count as well.

Since Putrix has twice as much fuel to burn as the Sun it will take it twice as long as the previous calculation. It also means that Putrix will release twice as much energy as the Sun will over its lifetime. So by the mathematical reasoning in part (b) E P = 2E and thus

We can mathematically represent this reasoning in the following way. Luminosity is energy per time so L ~ E/ , where E is the total amount of energy emitted by the star over its main sequence lifetime, and is the main sequence lifetime. The amount of energy emitted will be equal to the amount of energy produced by fusion of hydrogen to helium over that lifetime. E = mc 2 . Here m is the amount of matter that was converted into energy. As in the previous problem, 10% of the Sun's mass is available to undergo fusion, and 0.7% of that is converted into energy. So m = 0.007 x 0.1M . The same will hold for Putrix. So we can make a ratio

Luminosity is the amount of energy per second being given off by each star. It is therefore the rate at which energy is used up by the star. Since Putrix is 16 times more luminous that means that it is burning its fuel 16 times faster than the Sun. Since they have the same amount of fuel to burn (0.1M ) then Putrix will run out of fuel 16 times faster than the Sun.

When a cluster reaches 10 billion years old stars that are like the Sun begin to leave the main sequence and turn into Red Giants. There are no stars on the main sequence above it as they have all died as either supernovas or as Planetary Nebulae into White Dwarfs. Thus the main sequence terminates at the position of the Sun and there is a considerable Red Giant and White Dwarf population.

When a cluster has reached the age of 300 million years A stars are now running out of hydrogen in their cores and turning into Red Giants. All stars more massive than they (O and B) stars have long since turned into Red Giants and died as type II supernovae. The some of the A stars that are now turning into Red Giants are not massive enough to go supernova and will instead puff off their outer envelopes and end up as White Dwarfs. Remember that stars below 4M will turn into White Dwarfs. The A stars run from about 10 to 100 times the Luminosity of the Sun and since the Luminosity is proportional to mass to the fourth power that means the masses of the A stars are about 2M to 3.2M . So there should be a small population of White Dwarfs as well as a considerable Red Giant population visible.

When the cluster is only 2 million years old none of the stars are yet old enough to have left the Main Sequence. So there should be a full main sequence stretching fom M stars to O stars. Since no stars have left the main sequence yet, there should be no Red Giants or White Dwarfs.

Not all red giants are older than all main-sequence stars. Massive stars, born relatively recently, quickly evolve into red giants, while we know the Sun has been a main-sequence star for the last 5 billion years. As an example, a red giant of 10M could not be a very old star. Stars born with 10M will have luminosities of 10 4 L and thus consume their main-sequence fuel 10 4 /10 = 10 3 times faster than the Sun. Since the Sun's main-sequence lifetime is 10 10 years, the 10M star's lifetime will only be about 10 7 years. The red-giant phase is even shorter than the main-sequence phase --- say, 2 million years in this example. Thus, the 10M red giant has been a star for only about 12 million years. Compare that to the 5 billion years of the Sun's happy time on the main sequence.

When Stars Explode

Type I supernovae are observed to have no hydrogen lines visible in their spectra. They are seen in all kinds of galaxies. They arise when a white dwarf star is accreting matter from a binary companion which is overflowing its Roche lobe. If the white dwarf mass exceeds 1.4M (the Chandrasekhar limit), the star can no longer be supported by electron degeneracy pressure. It starts to collapse and a runaway chain of nuclear reactions occurs. Lower mass nuclei, such as carbon and oxygen, are fused into heavier ones, such as nickel and iron. A vast amount of energy is released which causes the star to explode and leave no remnant star. Material, enriched with heavy elements, is flung into space.

Type II supernovae are observed to have hydrogen lines visible in their spectra. They are seen primarily in the arms of spiral galaxies (where young stars live). These kinds of supernovae are the endpoint of evolution for very massive stars (M > 10M ). After the helium fusion process ends in the cores of these stars they have enough mass to ignite carbon fusion and fusion of heavier elements. An onion layered effect of fusion shells fusing different elements builds up in the interior of the star. The fusion process stops at iron; fusion of iron does not yield more energy than is put into it. An iron core builds up. When the core reaches a mass greater than 1.4M it collapses under gravity and becomes a neutron star, forcing the electrons of atoms into the nuclei and combining with protons to become neutrons and releasing neutrinos. The envelope of the star falls down upon the neutron star and rebounds off in an enormous explosion. Most of the energy released is in the form of neutrinos. Heavy elements, including some heavier than iron, are created during the explosion and flung out into interstellar space.