Question

Estimate the lifetime of sun whether your source was:

a) Potential gravitational

b)Energy of nuclear fusion of hidrogen and helium on PP cicle

c)Calculate the minimal mass of a star with lifetime equal to universe's age, or 13.7 billions of years

Answer 1

The gravitational energy can be treated as fuel for the Sun's motion. As the lifetime of Sun is obtained by considering that how fast the fuel consumes and amount of fuel Sun contains.

a) When Source is potential gravitational

The mass of Sun is

= 2 x 1030 kg

Gravitational can generate the energy and the gravitational energy = $\frac{3}{5}\times GM^{2}\div R$

radius of the Sun is approximately, R=700 million meter

therefore gravitational energy of the sun =$2.3\times 10^{41}$ joules

Now the lifetime =(Amount of fuel)/(fuel consumption per second),

i.e., lifetime =($2.3\times 10^{41}$ joules)$\div (3.8\times 10^{26}$ joules per second)

=0.6*10^15 seconds.

or = 20 million years

b) Energy of nuclear fision on p-p chain

$1_{H} +1_{H}\rightarrow 2_{H} +e^{+} +\nu _{e}$

$2_{H} +1_{H}\rightarrow 3_{He} + \gamma$

$3_{He} +3_{He}\rightarrow 4_{He} + 1_{H} + 1_{H}$

in these four protons are fused into one Heliom-4 and produce energy,

$1_{H} +1_{H} + 1_{H} + 1_{H}\rightarrow 4_{He} + Binding Energy$

because mass of one proton is $1.6726\times 10^{-27} kg$ andthe mass of one helium-4 is $6.643\times 10^{-27} kg$ , then

Binding energy = $4\times 1.6726\times 10^{-27} kg-6.643\times 10^{-27} kg$

or BE = $4.4\times 10^{-29} kg$

c) Mass of star-

as we know that,

$\frac{t_{MS}}{t_{S}}\sim \left ( \frac{M}{M_{S}} \right )^{-2.5}$

where $t_{MS}$ is solar lifetime, M is mass of star, $t_{S}$ is sun's lifetime and $M_{S}$ is the solar mass, therefore

$\frac{13.7 billion year}{20 million year}=$$\left ( \frac{M}{2\times 10^{30}kg} \right )^{-2.5}$

or $M=12.3\times 10^{6}\times 2\times \times 10^{30} kg$

or $M=24.6\times 10^{36} kg$