Weighing the Invisible

Show explicit units, and run a sanity check on every result. Worked solutions are released after the homework due date.

Useful constants (CGS):

Constant	Value
$G$	$6.674 \times 10^{-8}~\mathrm{cm^3\,g^{-1}\,s^{-2}}$
$M_\odot$	$2.0 \times 10^{33}~\mathrm{g}$
$1~\mathrm{kpc}$	$3.086 \times 10^{21}~\mathrm{cm}$
$1~\mathrm{AU}$	$1.496 \times 10^{13}~\mathrm{cm}$
$1~\mathrm{km\,s^{-1}}$	$10^{5}~\mathrm{cm\,s^{-1}}$
$1~\mathrm{yr}$	$3.16 \times 10^{7}~\mathrm{s}$

Useful relations: enclosed mass $M(\lt r) = rv^2/G$ ; a flat rotation curve has $v \approx \text{const}$ , so $M(\lt r) \propto r$ ; a central point mass gives $v \propto r^{-1/2}$ ; a circular orbit has $v = 2\pi a/P$ .

Conceptual

Problem

⭐⭐ Weighing Sagittarius A* — the inference. Stars at the Galactic center trace small, fast, closed orbits around an invisible point. The enclosed mass comes out near $4\times10^{6}\,M_\odot$ , packed inside the orbit of the closest star.

(a) State the observable, the model, and the inference for this measurement.
(b) Explain why the compactness — not the mass alone — is what forces the black-hole interpretation.
(c) Name one independent observation that supports the same conclusion.

Problem

⭐⭐ Flat does not mean empty. A student claims: “A flat rotation curve means the enclosed mass stops growing past the bright disk — the galaxy is basically empty out there.”

(a) Use $M(\lt r) = rv^2/G$ to show what a flat $v(r)$ actually implies for $M(\lt r)$ .
(b) Identify the student’s error in one sentence.
(c) Rising enclosed mass with no rising light tells you what must be present?

Problem

⭐ “Dark” is not “mysterious.” In two or three sentences total:

(a) State precisely what astronomers mean by calling dark matter “dark.”
(b) Name the kind of evidence that establishes it, given that no telescope images it directly.

Calculation

Problem

⭐⭐ The luminous-versus-dynamical ledger. A spiral galaxy has a flat rotation speed $v = 180~\mathrm{km\,s^{-1}}$ out to $r = 25~\mathrm{kpc}$ . Its stars and gas (its baryons) sum to $M_\text{bary} = 4\times10^{10}\,M_\odot$ .

(a) Compute the dynamical mass $M(\lt 25~\mathrm{kpc}) = rv^2/G$ in solar masses.
(b) Subtract the baryons to get $M_\text{dark}$ , and form the ratio $M_\text{dark}/M_\text{bary}$ .
(c) Inside this radius, is there more dark or baryonic matter — and by roughly what factor?

Problem

⭐⭐ What the Solar System would predict. Suppose a galaxy’s mass were all concentrated at its center, like the Sun in the Solar System. At $r_1 = 8~\mathrm{kpc}$ the orbital speed is $v_1 = 220~\mathrm{km\,s^{-1}}$ .

(a) For a central point mass, $v \propto r^{-1/2}$ . Predict the speed at $r_2 = 32~\mathrm{kpc}$ .
(b) The galaxy’s observed speed at $32~\mathrm{kpc}$ is still about $220~\mathrm{km\,s^{-1}}$ (a flat curve). Compare to your prediction.
(c) What does the gap between the Keplerian prediction and the flat observation reveal?

Problem

⭐⭐ Weigh the black hole with Kepler. The star S2 orbits Sagittarius A* with semi-major axis $a \approx 1000~\mathrm{AU}$ and period $P \approx 16~\mathrm{yr}$ . Treat the orbit as circular.

(a) Find the orbital speed $v = 2\pi a/P$ in $\mathrm{cm\,s^{-1}}$ .
(b) Use $M = rv^2/G$ with $r = a$ to estimate the enclosed mass in solar masses.
(c) This is the same physics that weighed binary stars in Module 2 — Kepler’s third law. The real S2 orbit is highly elliptical ( $e \approx 0.88$ ). Explain why the circular estimate is approximate, and whether using pericenter speed and distance would over- or under-estimate the mass.

Synthesis

Problem

⭐⭐ Two independent witnesses. Dark matter is a convergent inference. Build two distinct observe → model → infer chains for it, drawing on two different lines of evidence (galaxy rotation curves, the Bullet Cluster, cluster gravitational lensing, or the cosmic web).

(a) Chain 1: observable → model → inference.
(b) Chain 2: a different observable → model → inference.
(c) Why is agreement between two independent methods stronger than either alone?

Problem

⭐⭐ A dark engine with visible consequences. A supermassive black hole emits no light from inside its horizon, yet it can reshape a galaxy’s future star formation.

(a) Explain how an actively accreting black hole (an AGN) injects energy into the galaxy’s gas.
(b) Connect this to the ecosystem idea from the previous reading: how does AGN feedback resemble stellar feedback?
(c) Why can a galaxy’s center influence star formation across the whole galaxy?

Problem

⭐⭐⭐ Why the Bullet Cluster is decisive. A skeptic proposes: “There is no dark matter — we just haven’t counted all the ordinary gas, which is most of a cluster’s normal matter.”

(a) In the Bullet Cluster, where does most of the ordinary matter sit, and how is it located observationally?
(b) Where does the total gravitating mass sit, and how is it located?
(c) Explain why the spatial offset between those two defeats the skeptic — what would have to be true for “just more gas” to work, and why the collision rules it out.