The Sun, and General Properties of Stars
The Sun and General Properties of Stars

Answer the following questions on your own paper. Please show all work and answer each question completely.

It was once thought that the source of the Sun's energy was gravitational contraction, a process that we know generates heat even today (that's how protostars initially ignite their nuclear fuel). We know that the Sun must generate energy as fast as it releases it. The total energy released by the Sun per second is about 4 x 1033 ergs. So if we can just find out how much energy the Sun can release over its whole lifetime (in ergs), we can determine how long it will live.

A similar process can be used for a car. Say a car uses up 2 gallons of gas per hour. If we figure out how much fuel the car can hold (say, 20 gallons), then we know the car's lifetime of burning 2 gallons of fuel every hour will be 10 hours.

(1) If we assume the Sun gets all of its energy from gravitational contraction, we find that the Sun has 4 x 1048 ergs of available "fuel''. How long can the Sun live (in years) with gravity as its only source of energy?
This result was widely known and accepted around the turn of the century, until geologists using radioactive dating determined that the age of the Earth was about 5x109 years (5 billion years) old, much older than the apparent age for the Sun estimated in (1)! In fact, this was used as a creationist argument, supporting the fact that the Earth is indeed the oldest object in the Universe and hence must have been created.

Much later, when the theory of nuclear fusion was being developed, scientists applied their theory to stellar energy generation and the Sun. Einstein's famous E = mc2 equation doesn't quite work here, since the reaction involved only converts a small fraction of the mass involved into useful energy. Still, there's a lot of energy involved.
(2) Assuming that 10% of the Sun's mass is available for nuclear energy generation during its main sequence lifetime (only the Hydrogen close to the core, for reasons we will get into in class), and assuming that the mass is converted into energy with an efficiency of only 0.7%

E = 0.1*0.007*MSunc2,

calculate the total energy (in ergs) of the Sun. The mass of the Sun is 2 x 1033 grams and the speed of light is 3 x 1010 cm/sec.

(3) Now estimate the lifetime of the Sun in years as in (1) with nuclear fusion as the power source. Your answer should be more in line (in order of magnitude at least) with the estimated age of the Earth.

(4) We said in problem (2) that only the Hydrogen close to the core of the Sun or any other star will participate in the nuclear energy generation of the star during its main sequence lifetime. Explain why this is true.

Atomic reactors (and weapons) produce large amounts of energy. Below you will compare this energy to the energy that a star gives off when it explodes as a supernova. (We will discuss supernova later in the quarter, for now it is sufficient to know that stars more massive than our Sun end their fusion lifetimes with a supernova event.)

(5) When it first ignites, a rough estimate for the luminosity of a supernova is 1051 ergs/sec (about 1018 or a billion billion times more luminous than the Sun). We are located at about 1013 cm from the Sun. Imagine a supernova explodes at the center of the Solar System. We want to compare the amount of energy we receive from a supernova explosion to the energy we receive (remember the inverse square goes as 1/r2) at a distance of about 1 km from an atomic blast (1 km = 105) cm). A modern atomic weapon is rated at 50 Megatons (that's the amount of energy it can release), which is roughly 1024) ergs released per second. What is the ratio of the energy received from a supernova explosion on the Earth from the Sun to the ratio of energy we'd receive viewing a nuclear explosion from 1 km away?
The questions below will help you understand parallax, and absolute versus apparent brightness.
(6) There is a certain class of star called r-Lisam and all of these type of stars have the same luminosity (absolute brightness) = 3 x 1033) ergs/sec. One example of this type of star (Star A) has a parallax of 1/50 arcsecs. How far (in parsecs) is this star from Earth?

(7) There is a another r-Lisam star (Star B) that has a parallax of 1/50 arcsecs as seen from SATURN. How far (in parsecs) is this star from Saturn? (Useful information: Saturn is 10 AU from the Sun.) About how far (in parsecs) is this star from Earth?

(8) The apparent brightness of Star A (as measured in those funny units again) is 6x1032 ergs/sec/cm-2. What is the apparent brightness of Star B as seen from Earth? What would be the absolute brightness of Star B as observed from Neptune?