The Lifetime of the Sun

The sun gives off energy all of the time. This is the energy that all life uses to grow and live; whether directly (as photosynthesis by plants) or indirectly (as herbivores and carnivores that consume the energy stored in living things). Without that energy source, the Earth would be a dark, cold, lifeless place. So, it is certainly of interest to ask "Will the Sun be there tomorrow?", or, more usefully, to ask when the sun will stop shining. In other words, how long will the sun last?
The goal of this homework assignment is to answer that question. The Sun formed from a spinning cloud of gas through gravitational collapse. The planets formed at the same time. So the ages of the planets provide a good estimate of the age of the Sun: 4.5 billion years. The Sun has been shining brightly, at almost exactly the same rate (a constant luminosity), for 4.5 billion years. This implies that it has been producing energy at a constant rate for those 4.5 billion years.

  1. Why can we say that the rate at which the sun gives off energy at the surface (the luminosity) must be equal to the rate at which it produces energy deep down in the core? (Hint: What would happen to the temperature of the sun if the two rates werent the same?)

    So, measuring the luminosity of the sun is equivalent to measuring the rate at which the nuclear reactions produce energy in the core of the sun. But we think we know exactly how those reactions work:

    4 H atoms -> 1 He atom + energy

    where the energy is released because some of the mass of the Hydrogen atoms is converted to energy.

    Mass of 1 Hydrogen atom:       1.673 x 10-24 grams
    Mass of 1 Helium atom:         6.644 x 10-24 grams
  2. This nuclear reaction has an input (4 H atoms) and two outputs : 1 He atom and energy. Since the He atom has less mass than 4 H atoms, that difference in mass must have been converted to energy. If the energy is produced via E=mc2, how much energy is produced by producing one He atom from 4 H atoms? (see also the units notes at the bottom). (Hint: How much mass is converted into energy?)

    So, for every 4 hydrogen atoms fused, we get the amount of energy you calculated above. But we know how fast the sun has produced energy (the luminosity, according to #1), so we know how fast the hydrogen fuel in the core of the sun is being used up.

  3. If the sun gives off 3.89 x 1033 ergs every second, how many hydrogen atoms are being destroyed every second?

    The sun will remain a main sequence star until it runs out of hydrogen fuel in the core of the star. The core of the sun contains about 10% of the total mass of the star.

  4. Why are these reactions confined to the core?

  5. The total mass of the sun is 2x1033 gm. How long will it take until the sun has used up all the hydrogen atoms in the core (the central 10%)? That is, what is the main-sequence lifetime of the sun?

  6. Compare the lifetime of the sun to the current age. How soon will the sun running out of fuel be a problem?

Units Notes:
Luminosity - energy per second; usually the total energy given off by an object per second.
1 Watt = 107 ergs/sec
1 Solar Luminosity = 3.89 x 1033 ergs/sec

Energy - Bah. Try and define that. Usually measured in ergs; an erg is about the energy of one flea jump.
1 erg = 1 gm*cm2/sec2 = 10-7 joules
[if you use E=mc2 with the mass in grams, the speed in cm/sec, then you get E in gm*(cm/sec)2, which is gm*cm2/sec2 = ergs]

Mass - the amount of matter. Usually measured in grams, or solar masses.
1 solar mass = 1.989 x 1033 gm
1 hydrogen atom = 1 proton = 1.67352 x 10-24 gm

Speed - velocity; distance traveled per time unit. Measured in lots of units; We'll use cm/sec because of the definition energy in ergs
Speed of light = c = 3.00 x 1010 cm/sec = 300,000 km/sec

Time - measured in seconds, days, months, years.
1 year = 3.15 x 107 seconds