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Galaxy distance questions

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The Andromeda galaxy, also known as M31, is the nearest spiral galaxy to the Milky Way and the most distant object visible to the naked eye in the night sky. Andromeda is at a distance of 660 kpc from the Milky Way and it is moving towards the Milky Way at about 120 km s-1, with a collision and merger likely in about 4.5 billion years.

(a) (i) Combine Newton's second law of motion and Newton's law of gravity for the force between two masses m1 and m2, separated by a distance r, to show that the acceleration of mass m1 as a result of the gravitational force may be written as

a=G m2 where G=6.7x10-11 Nm2 Kg-2

Second law of motion= F=ma

Show your working clearly at each stage.

(ii) The mass of the Andromeda galaxy has been estimated as 7.1 1011 solar masses (take one solar mass to be 2.0 1030 kg). Assuming that the Milky Way has the same mass as the Andromeda galaxy and ignoring other galaxies, find the magnitude and direction of the acceleration of the Andromeda galaxy, showing your working. State whether this acceleration will increase or decrease in future, giving a brief reason.

(b) (i) The motion of the Andromeda galaxy relative to the Milky Way can be determined by measuring the wavelength of known spectral absorption lines. Find the wavelength of a line that would be observed in a spectrum from Andromeda if the same line had a wavelength of 500.00 nm in a reference spectrum.

(ii) The Hubble relationship states that the apparent speed with which a distant galaxy is moving away from us is v=Hor
where H0 = 70 km s-1 Mpc-1. Find the wavelength the same line would have if the Andromeda galaxy obeyed the Hubble relationship.

(iii) Briefly explain why your answers to part (b)(i) and (ii) differ. (We expect that you will be able to answer this in two or three sentences.)

(c) (i) Hubble's observations of distant galaxies, published in 1929, gave a value of H0 = 500 km s-1 Mpc-1. Since then, Hubble's value has been revised downwards by more accurate observations. Briefly explain why the value of H0 is linked to an estimate of the age of the Universe and what the revisions to Hubble's value of H0 imply for estimates of the age of the Universe derived from them, compared with the original value.
(We expect that you will be able to answer this in up to 100 words.)

(ii) Using the fact that the value of H0 given in part (b)(ii) of the question implies that the Universe began about 14 billion years ago, explain why Hubble's original value was not accepted, given that radiometric dating of some meteorites implies ages of about 4.5 billion years. (We expect that you will be able to answer this in up to 100 words.)

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Solution Summary

We solve problems in cosmology involving Hubble's law and the Andromeda galaxy.

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Please see the attachment.

(a) i. The gravitational force of attraction acting on due to is given by

By Newton's second law, we also have

Thus we have

Dividing both sides by yields

ii. The gravitational acceleration of the Andromeda galaxy toward the Milky Way is given by

Since the Andromeda galaxy is currently moving toward the Milky Way, ...

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