Question

If the dark matter in the Milky Way were composed entirely of MACHOs (evidence shows it is not), approximately how many would there have to be? Assume the average mass of a MACHO is\({\rm{1/1000}}\)that of the Sun, and that dark matter has a mass\({\rm{10}}\)times that of the luminous Milky Way galaxy with its\({\rm{1}}{{\rm{0}}^{{\rm{11}}}}\)stars of average mass\({\rm{1}}{\rm{.5}}\)times the Sun’s mass.

Short answer

The number of MACHOS is obtained as: \({\rm{1}}{\rm{.5 \times 1}}{{\rm{0}}^{{\rm{15}}}}\).

Step-by-step solution

Expression for the number of MACHOs

The expression for the number of MACHOs is given by,

\(n = \frac{{{m_{dm}}}}{{{m_M}}}\)

Here\({m_{dm}}\)is the total mass of the dark,\({m_M}\)is the mass of a MACHO.

Evaluating the number of MACHOs

If the dark matter in our galaxy were composed entirely based upon the MACHOs,

Then, it has \({\rm{1/1000}}\) the mass of our Sun.

So, the number of MACHOS would be:

\(\begin{align}N &= \frac{{{\rm{Total mass of dark matter}}}}{{{\rm{mass per MACHO}}}}\\ &= \frac{{{\rm{10 \times 1}}{{\rm{0}}^{{\rm{11}}}}{\rm{ \times 1}}{\rm{.5 \times }}{{\rm{m}}_{{\rm{sun}}}}}}{{\frac{{{{\rm{m}}_{{\rm{sun}}}}}}{{{\rm{1000}}}}}}\\ &= {\rm{1}}{\rm{.5 \times 1}}{{\rm{0}}^{{\rm{15}}}}\end{align}\)

Therefore, the number of MACHOs that make up the dark matter in Milky galaxy is \({\rm{1}}{\rm{.5 \times 1}}{{\rm{0}}^{{\rm{15}}}}\).