Question

(a) Estimate the mass of the luminous matter in the known universe, given there are\({\rm{1}}{{\rm{0}}^{{\rm{11}}}}\)galaxies, each containing\({\rm{1}}{{\rm{0}}^{{\rm{11}}}}\)stars of average mass\({\rm{1}}{\rm{.5}}\)times that of our Sun. (b) How many protons (the most abundant nuclide) are there in this mass? (c) Estimate the total number of particles in the observable universe by multiplying the answer to (b) by two, since there is an electron for each proton, and then by\({\rm{1}}{{\rm{0}}^{\rm{9}}}\), since there are far more particles (such as photons and neutrinos) in space than in luminous matter.

Short answer

(a) The mass is obtained as: \({M_{tot}} = {\rm{ 3}}{\rm{.24 x 1}}{{\rm{0}}^{{\rm{52}}}}{\rm{ kg}}\).

(b) The number of protons is obtained as: \({\rm{1}}{\rm{.87 \times 1}}{{\rm{0}}^{{\rm{79}}}}\).

(c) The total number of particles is obtained as: \({\rm{3}}{\rm{.6 \times 1}}{{\rm{0}}^{{\rm{88}}}}\).

Step-by-step solution

Mass of the luminous matter.

The approximate mass of the luminous matter in each galaxy is given by,

\({M_L} = n \times 1.5{m_s}\)

Here\(n\)is the number of stars in the galaxy and\({m_s}\)is the mass of the sun.

Evaluating the mass

From the given data,

The mass is obtained as:

\(\begin{array}{c}m = {\rm{ number of stars \times average star mass}}\\ = {\rm{ 1}}{{\rm{0}}^{{\rm{11}}}}{\rm{ \times 1}}{{\rm{0}}^{{\rm{11}}}}{\rm{ \times 1}}{\rm{.5 \times }}{m_{Sun}}\\ = {\rm{ 1}}{{\rm{0}}^{{\rm{22}}}}{\rm{ \times 1}}{\rm{.5 \times 2 \times 1}}{{\rm{0}}^{{\rm{30}}}}{\rm{ kg}}\\ = {\rm{ 3}}{\rm{.0 \times 1}}{{\rm{0}}^{{\rm{52}}}}{\rm{ kg}}\end{array}\)

Therefore, the mass is: \({\rm{3}}{\rm{.0 \times 1}}{{\rm{0}}^{{\rm{52}}}}{\rm{ kg}}\).

Evaluating the number of protons

The mass of the proton is:

\({m_p}{\rm{ = 1}}{\rm{.67 \times 1}}{{\rm{0}}^{{\rm{ - 27}}}}{\rm{ kg}}\)

Then, it is evaluated as:

\(\begin{array}{c}{n_p} &= \frac{m}{{{m_p}}}\\ &= \frac{{{\rm{3}}{\rm{.0 \times 1}}{{\rm{0}}^{{\rm{52}}}}{\rm{ kg}}}}{{{\rm{1}}{\rm{.67 \times 1}}{{\rm{0}}^{{\rm{ - 27}}}}{\rm{ kg}}}}\\ &= {\rm{1}}{\rm{.79 \times 1}}{{\rm{0}}^{{\rm{79}}}}\end{array}\)

Therefore, the protons are: \({\rm{1}}{\rm{.79 \times 1}}{{\rm{0}}^{{\rm{79}}}}\).

Evaluating the total number of particles

The total number of particles in the observable universe is:

\(\begin{array}{c}{n_{tot}} &= {\rm{ }}{{\rm{n}}_{\rm{p}}}{\rm{ \times 2 \times 1}}{{\rm{0}}^{\rm{9}}}\\ & = 1.87 \times 1{0^{79}} \times 2 \times 1{0^9}\\ & = {\rm{ 3}}{\rm{.74 \times 1}}{{\rm{0}}^{{\rm{88}}}}\end{array}\)

Therefore, the total number of particles is: \({\rm{3}}{\rm{.74 \times 1}}{{\rm{0}}^{{\rm{88}}}}\).