2. Martin sees Sally's clock run slow, because he perceives himself to be at rest and Sally to be moving. BUT -- from Sally's point of view, she is at rest and Martin is moving. So Sally will see Martin's clock running slow as well.
(That's what it means to give up the notions of "absolute space" and "absolute time" -- there are no objective answers to the questions, "Which of them is moving?" and "Whose clock is running slower?" The answers to those questions depend completely on your frame of reference.)
6. If a gravitationally collapsing object (like the core of a star) has a mass of more than around 3 solar masses, then its collapse cannot be halted by either degenerate electron pressure or degenerate neutron pressure, because neither of those pressures are strong enough. So as long as a high-mass star has a core with a mass of more than 3 solar masses, that core will always become a black hole singularity.
9. As we said in class, if a black hole is orbiting another star, then there can be an accretion disk around the black hole which gives off copious amounts of X-rays (but very little visible light).
The problem is, if the normal star's companion was a white dwarf or a neutron star, there could also be an accretion disk around the companion that was hot enough to give off X-rays (because both white dwarfs and neutron stars have quite strong gravity). Since white dwarfs and neutron stars are so faint, it would be difficult to see them with visible light next to the normal star.
So: we have a normal star, and a not-visible companion that emits X-rays. How do we know that the non-visible companion is a black hole rather than a white dwarf or a neutron star? We must observe the orbit of the normal star around its dim companion, and use Newton's version of Kepler's Third Law to determine the mass of the mystery object. If that mass is less than 3 solar masses, the object could be a white dwarf or neutron star. But if its mass is more than 3 solar masses, it can only be a black hole.
12. If the Sun collapsed into a black hole (which cannot happen, since its mass is less than 3 solar masses), the Earth's orbit would not be affected at all. The Earth's orbit is determined by the gravitational pull from the Sun, and that gravitational pull only depends on the mass of the Sun (which does not change when it becomes a black hole), the distance between the Earth and the Sun (which is still 1 AU), and the mass of the Earth (which of course is still the same). So the Sun-as-black-hole will pull on the Earth with exactly the same strength as it did when it was a normal star, and the Earth's orbit will remain unchanged.
17. A distant observer would see an object falling towards a black hole take an infinite amount of time just to reach the black hole's event horizon. In other words, we would never see the object fall past the event horizon into the black hole. This is because of gravitational time dilation -- clocks run slow in a strong gravitational field. So as the object approaches the event horizon, we will see time run slower and slower for that object, and the object will be seen to move more and more slowly.
26. The mass of the supermassive black hole at M87's center is given on page 551 as 3 billion solar masses. The mass of the star orbiting the black hole is not given, but whatever it is, it will be so small compared to the black hole's mass as to be negligible. The semimajor axis of the star's orbit around the black hole is given as 500 AU. So, using the version of Kepler's Law found on page 441, we find the period of the orbit to be:
To fully appreciate this, note that if the radius of the star's orbit around the black hole is (500 AU)(1.5 x 108 km/AU) = 7.5 x 1010 km, then its circumference (the distance the star travels in one orbit around the black hole) is 2 pi times this: 4.7 x 1011 km. The star completes one orbit in just (0.2 years)(365 days/year) = 73 days, which is equal to (73 days)(24 hours/day)(60 minutes/hour)(60 seconds/minute) = 6.3 x 106 seconds. So its speed must be (4.7 x 1011 km) / (6.3 x 106 s) = 75,000 km/s! This is one-fourth of the speed of light (or about 170 million mph)!
27. According to page A-1 of your textbook, the mass of Jupiter is 1.899 x 1027 kg. Using the formula for the Schwarzschild radius given on page 553, we find:
In other words, if you could somehow compress Jupiter (which currently has a radius of 71,500,000 meters) into a sphere with radius of just 2.81 meters, then Jupiter would become a black hole.
Last edited 18 Apr 04 M. A. Weinstein.