Astronomy
1st Edition
ISBN: 9781938168284
Author: Andrew Fraknoi; David Morrison; Sidney C. Wolff
Publisher: OpenStax
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Chapter 24, Problem 20E
Look up G, c, and the mass of the Sun in Appendix E and calculate the radius of a black hole that has the same mass as the Sun. (Note that this is only a theoretical calculation. The Sun does not have enough mass to become a black hole.)
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The Schwarzschild radius RBH for an object of mass M is defined as
(See image.)
where c is the speed of light and G is the universal gravitational constant. RBH gives the radius of the event horizon of a black hole with mass M. In other words, it gives the radius to which some amount of mass M would need to be compressed in order to form a black hole.
1. The mass of the Sun is about 1.99 × 1030 kg. What would be the radius of a black hole with this mass?
2. The mass of Mars is about 6.42 × 1023 kg. What would be the radius of a black hole with this mass?
3. Suppose you want to make a black hole that is roughly the size of an atom (take RBH = 1.10 x 10-10 m). What would be the mass M of such a black hole?
In 1999, scientists discovered a new class of black holes with masses 100 to 10,000 times the mass of our sun that occupy less space than our moon. Suppose that one of these black holes has a mass of 1×1021×102 suns and a radius equal to one-half the radius of our moon.
A)What is the density of the black hole in g/cm3g/cm3? The radius of our sun is 7.0×105km7.0×105km, and it has an average density of 1.4×103kg/m31.4×103kg/m3. The diameter of the moon is 2.16×1032.16×103 miles. 1km=0.6214mile1km=0.6214mile.?
The radius Rh of a black hole is the radius of a mathematicalsphere, called the event horizon, that is centered on the blackhole. Information from events inside the event horizon cannotreach the outside world. According to Einstein’s general theory ofrelativity, Rh = 2GM/c2, where M is the mass of the black hole andc is the speed of light.Suppose that you wish to study a black hole near it, at a radialdistance of 50Rh. However, you do not want the difference in gravitationalacceleration between your feet and your head to exceed10 m/s2 when you are feet down (or head down) toward the blackhole. (a) As a multiple of our Sun’s mass MS, approximately what isthe limit to the mass of the black hole you can tolerate at the givenradial distance? (You need to estimate your height.) (b) Is the limitan upper limit (you can tolerate smaller masses) or a lower limit(you can tolerate larger masses)?
Chapter 24 Solutions
Astronomy
Ch. 24 - How does the equivalence principle lead us to...Ch. 24 - If general relativity offers the best description...Ch. 24 - Einstein’s general theory of relativity made or...Ch. 24 - If a black hole itself emits no radiation, what...Ch. 24 - What characteristics must a binary star have to be...Ch. 24 - A student becomes so excited by the whole idea of...Ch. 24 - What is an event horizon? Does our Sun have an...Ch. 24 - What is a gravitational wave and why was it so...Ch. 24 - What are some strong sources of gravitational...Ch. 24 - Suppose the amount of mass in a black hole...
Ch. 24 - Imagine that you have built a large room around...Ch. 24 - A monkey hanging from a tree branch sees a hunter...Ch. 24 - Why would we not expect to detect X-rays from a...Ch. 24 - Look elsewhere in this book for necessary data,...Ch. 24 - Which is likely to be more common in our Galaxy:...Ch. 24 - If the Sun could suddenly collapse to a black...Ch. 24 - Suppose the people in Figure 24.4 are in an...Ch. 24 - You arrange to meet a friend at 5:00 p.m. on...Ch. 24 - You are standing on a scale in an elevator when...Ch. 24 - Look up G, c, and the mass of the Sun in Appendix...Ch. 24 - Suppose you wanted to know the size of black holes...Ch. 24 - Use the result from Exercise 24.21 to calculate...Ch. 24 - Since the force of gravity a significant distance...
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- In 1999, scientists discovered a new class of black holes with masses 100 to 10000 times the mass of our sun, but occupying less space than our moon. Suppose that one of these black holes has a mass of 1x10^3 suns and a radius equal to one-half the radius of our moon. What is the density of the black hole in g/cm^3? The radius of our sun is 7.0x10^5km and it has an average density of 1.4x10^3kg/m^3. The diameter of the moon is 2.16x10^3 miles. Note: the volume of a sphere is V=4/3 pie r^3arrow_forwardThe Schwarzschild radius, RS, of a black hole depends on its mass m, the speed of light c,and the gravitational constant G (with units m3/(kg s2)).Find a dimensionally correct expression for RS in terms of these quantities.Assuming that you found a dimensionally correct expression, can you be sure that this expression is, in fact, the correct expression for calculating the Schwarzschild radius? Explainyour answer.arrow_forwardIn 1999 scientists discovered a new class of black holes with masses 100 to 10,000 times the mass of our sun but occupying less space than our moon. Suppose that of these black holes has a mass of 1x10^3 sun's and radius equal to one-half the radius of our moon. What is the density in grams per cubic centimeter? The mass of the sun is 2.0x10^30 kg and the radius of the moon is 2.16x10^3 mi.arrow_forward
- Assume the event horizon is the size of the black hole (it is not: in reality, the black hole is a point source). What is the density of a black hole that has the mass of the Sun in units of kg/cm3? Please express your answer in scientific notation with the "e" format inside the text box, and keep three significant figures (for example, 5.01e3 for 5.01 × 103).arrow_forwardThe following quotation is taken from the article “Quantum Black Holes”, by Bernard J. Carr and Steven B. Giddings, in the May 2005 issue of Scientific American. "The total time for a black hole to evaporate away is proportional to the cube of its initial mass. For a solar-mass hole, the lifetime is an unobservably long 1064 years." a. Recall that the solar mass is 2 × 10³0 kilograms. Write a formula for the lifetime, L, of a black hole as a function of its mass, m. Start by finding the value of the constant k, then write your function using the letter k (rather than its value in scientific notation). For example, for a direct variation you would write “L(m) = km”. k = a × 10¹ where a = L(m) = b. The present age mass = c × 10ª kg, where c = A and b = = of the universe is about 10¹0 years. What would be the mass of a black hole as old as the universe? ID and d = Jarrow_forwardIn 1999, scientists discovered a new class of black holes with masses 100 to 10,000 times the mass of our sun that occupy less space than our moon. Suppose that one of these black holes has a mass of 1x10^3 suns and a radius equal to one-half the radius of our moon. What is the density of the black hole in g/cm^3? The radius of our sun is 7.0x10^5 km, and it has an average density of 1.4x10^3 kg/m^3. The diameter of the moon is 2.16x10^3 miles.arrow_forward
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