Final answer:
Based on the assumption that dark matter has a mass 10 times that of the Milky Way's luminous matter, and given the average mass of a MACHO is 1/1000 of the Sun's mass, the correct answer is that there would have to be approximately 10²³ MACHOs if they composed all dark matter in the galaxy.
Step-by-step explanation:
To find out approximately how many MACHOs would compose the dark matter in the Milky Way if all dark matter were in this form, we first need to know the total mass of dark matter and the mass of a typical MACHO. It has been given that the luminous mass of the Milky Way is made up of roughly 10¹¹ stars of average mass 1.5 times that of our Sun. Therefore, the total mass of luminous matter would be 10¹¹ stars × 1.5 solar masses per star. Since it is stated that dark matter has a mass 10 times that of the luminous matter, we would need to calculate the mass of 10¹¹ stars × 1.5 solar masses and then multiply that by 10 to get the total dark matter mass.
Once we have the total dark matter mass, we would divide this by the mass of an average MACHO, which is 1/1000 of the Sun's mass, to get the number of MACHOs needed. With these calculations, we can determine which of the provided options (a) 10¹´, (b) 10¹·, (c) 10²°, or (d) 10²³ is the correct number of MACHOs.
Performing the calculation, we get the total luminous mass as 1.5 × 10¹¹ solar masses. Multiplying this by 10 gives the total dark matter mass as 15 × 10¹¹ solar masses. Dividing this by 1/1000 of the solar mass, we find the number of MACHOs as 15 × 10¹´. Hence, the correct answer is (d) 10²³ MACHOs.