Problems & Exercises

Find the approximate mass of the luminous matter in the Milky Way galaxy, given it has approximately ${\text{10}}^{\text{11}}$ stars of average mass 1.5 times that of our Sun.

(a) Estimate the mass of the luminous matter in the known universe, given there are ${\text{10}}^{\text{11}}$ galaxies, each containing ${\text{10}}^{\text{11}}$ stars of average mass 1.5 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 ${\text{10}}^9$, since there are far more particles (such as photons and neutrinos) in space than in luminous matter.

On average, how far away are galaxies that are moving away from us at 2.0% of the speed of light?

(a) What is the approximate speed relative to us of a galaxy near the edge of the known universe, some 10 Gly away? (b) What fraction of the speed of light is this? Note that we have observed galaxies moving away from us at greater than $0\text{.}9c$.

Assuming a circular orbit for the Sun about the center of the Milky Way galaxy, calculate its orbital speed using the following information: The mass of the galaxy is equivalent to a single mass $1\text{.}5\times {\text{10}}^{\text{11}}$ times that of the Sun (or $3\times {\text{10}}^{\text{41}}\text{kg}$), located 30,000 ly away.

Andromeda galaxy is the closest large galaxy and is visible to the naked eye. Estimate its brightness relative to the Sun, assuming it has luminosity ${\text{10}}^{\text{12}}$ times that of the Sun and lies 2 Mly away.

The average particle energy needed to observe unification of forces is estimated to be ${\text{10}}^{\text{19}}\text{GeV}$. (a) What is the rest mass in kilograms of a particle that has a rest mass of ${\text{10}}^{\text{19}}\text{GeV/}{c}^2$? (b) How many times the mass of a hydrogen atom is this?

(a) What Hubble constant corresponds to an approximate age of the universe of ${\text{10}}^{\text{10}}$ y? To get an approximate value, assume the expansion rate is constant and calculate the speed at which two galaxies must move apart to be separated by 1 Mly (present average galactic separation) in a time of ${\text{10}}^{\text{10}}$ y. (b) Similarly, what Hubble constant corresponds to a universe approximately $2\times {\text{10}}^{\text{10}}$-y old?

The core of a star collapses during a supernova, forming a neutron star. Angular momentum of the core is conserved, and so the neutron star spins rapidly. If the initial core radius is $5\text{.}0\times {\text{10}}^5\text{km}$ and it collapses to 10.0 km, find the neutron star’s angular velocity in revolutions per second, given the core’s angular velocity was originally 1 revolution per 30.0 days.

Distances to the nearest stars (up to 500 ly away) can be measured by a technique called parallax, as shown in Figure 34.26. What are the angles ${\theta }_1$ and ${\theta }_2$ relative to the plane of the Earth’s orbit for a star 4.0 ly directly above the Sun?

What is the Schwarzschild radius of a black hole that has a mass eight times that of our Sun? Note that stars must be more massive than the Sun to form black holes as a result of a supernova.

Supermassive black holes are thought to exist at the center of many galaxies.

(a) What is the radius of such an object if it has a mass of ${\text{10}}^9$ Suns?

(b) What is this radius in light years?

The characteristic length of entities in Superstring theory is approximately ${\text{10}}^{-\text{35}}\text{m}$.

(a) Find the energy in GeV of a photon of this wavelength.

(b) Compare this with the average particle energy of ${\text{10}}^{\text{19}}\text{GeV}$ needed for unification of forces.

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 1/1000 that of the Sun, and that dark matter has a mass 10 times that of the luminous Milky Way galaxy with its ${\text{10}}^{\text{11}}$ stars of average mass 1.5 times the Sun’s mass.

Assume the average density of the universe is 0.1 of the critical density needed for closure. What is the average number of protons per cubic meter, assuming the universe is composed mostly of hydrogen?

A section of superconducting wire carries a current of 100 A and requires 1.00 L of liquid nitrogen per hour to keep it below its critical temperature. For it to be economically advantageous to use a superconducting wire, the cost of cooling the wire must be less than the cost of energy lost to heat in the wire. Assume that the cost of liquid nitrogen is $0.30 per liter, and that electric energy costs $0.10 per kW·h. What is the resistance of a normal wire that costs as much in wasted electric energy as the cost of liquid nitrogen for the superconductor?

Critical Thinking Assume two galaxies are $2.00\times {10}^2\text{ Mly}$ apart. (a) How fast are they separating due to space expansion? (b) Now suppose a third galaxy is located $2.00\times {10}^2\text{ Mly}$ along a line that is a perpendicular bisector of the line segment between the first two galaxies. How fast is this galaxy separating from either of the other two galaxies? (c) Why would the third galaxy appear to be receding from both of the initially mentioned galaxies at the same rate? (d) How would the apparent motion be detected?