The Final States

Show explicit units, and run a sanity check on every result. A scaling answer is judged by its exponents, not its coefficient. Worked solutions are released after the homework due date.

Useful constants (CGS):

Constant	Value
$G$	$6.674 \times 10^{-8}~\mathrm{cm^3\,g^{-1}\,s^{-2}}$
$c$	$3.0 \times 10^{10}~\mathrm{cm\,s^{-1}}$
$M_\odot$	$2.0 \times 10^{33}~\mathrm{g}$
$R_\odot$	$7.0 \times 10^{10}~\mathrm{cm}$
$1~\mathrm{km}$	$10^5~\mathrm{cm}$
Schwarzschild scaling	$R_s \approx 3.0~\mathrm{km}\,(M/M_\odot)$
Course TOV value	$M_\text{TOV} \approx 2.5\,M_\odot$ (order-of-magnitude divider)
Chandrasekhar mass	$M_\text{Ch} \approx 1.4\,M_\odot$

Useful relations from the reading: $R_{\max} = cP/(2\pi)$ (surface speed below $c$ ); compactness $R_s/R$ ; $v_\text{esc}/c = \sqrt{R_s/R}$ ; $P\propto R^2$ at fixed angular momentum.

Conceptual

Problem

⭐ Pulsars as evidence. A pulsar pulses every few milliseconds, with clock-like regularity.

(a) What is the observable in the pulsar argument?
(b) What compact-object model explains that observable?
(c) What physical inference does the model force about the source’s size and density?

Problem

⭐⭐ The TOV limit is not a second Chandrasekhar constant.

(a) Why is the Chandrasekhar limit more precisely known than the TOV limit?
(b) Name two pieces of physics that make the neutron-star maximum mass harder to calculate.
(c) Explain why a securely measured $2.1\,M_\odot$ neutron star is scientifically important.

Problem

⭐⭐ A boundary, not a wall. A student says: “The event horizon is where gravity finally gets strong enough to trap light — like an invisible wall around the black hole.”

(a) Why is “a wall” the wrong mental picture? (What actually happens to infalling matter at the horizon, locally?)
(b) The horizon is a boundary in spacetime, defined by causal structure. State what is true of every future-directed path once inside.
(c) A distant observer watching something fall in never sees it cross. Explain why — and why this is not evidence that the horizon is a solid barrier.

Calculation

Problem

⭐ Schwarzschild radius by scaling. A stellar-mass black hole has mass $8\,M_\odot$ .

(a) Use $R_s \approx 3.0~\mathrm{km}\,(M/M_\odot)$ to estimate its Schwarzschild radius.
(b) Convert your answer to centimeters.
(c) Compare your result to a neutron-star radius of $10~\mathrm{km}$ .

Problem

⭐⭐ Compactness and escape speed. A neutron star has mass $1.4\,M_\odot$ and radius $12~\mathrm{km}$ .

(a) Estimate $R_s$ using the solar-mass scaling.
(b) Compute the compactness $R_s/R$ .
(c) Use $v_\text{esc}/c = \sqrt{R_s/R}$ to estimate the escape speed as a fraction of $c$ .
(d) Explain why this object requires general relativity for accurate modeling.

Problem

⭐⭐ Pulse period as a size limit. A millisecond pulsar has period $P = 2.0~\mathrm{ms}$ .

(a) Convert the period to seconds.
(b) Use $R_{\max} = cP/(2\pi)$ (from requiring the surface speed to stay below $c$ ) to estimate the largest possible radius.
(c) Convert your answer from centimeters to kilometers.
(d) Explain why this rules out a normal star.

Synthesis

Problem

⭐⭐ Classifying an unseen compact object. A binary holds an unseen companion of mass $5\,M_\odot$ , emitting strong X-rays from accreting gas, with no evidence for thermonuclear bursts from a surface.

(a) Why is a white-dwarf model unlikely?
(b) Why is a neutron-star model unlikely?
(c) What compact-object model is favored?
(d) State the assumption about the TOV limit that enters your inference.

Problem

⭐⭐ Converging evidence. Choose two of: pulsar pulses, X-ray binaries, stellar orbits around Sgr A*, gravitational waves, Event Horizon Telescope images.

(a) For each chosen observation, identify the observable.
(b) For each, identify the physical model.
(c) Explain how the two together make the case for compact remnants stronger than either alone.

Problem

⭐⭐⭐ Initial mass is not destiny. Two stars both begin near $25\,M_\odot$ ; one leaves a neutron star, the other a black hole.

(a) Explain why this is possible even though a table gives $\sim 25\,M_\odot$ as an approximate boundary.
(b) Name three physical effects that can shift the outcome.
(c) In a short paragraph, explain why final core mass predicts the remnant better than initial mass alone.