USAAAO 2026, I кръг — Задача 3

• D

Solution: Given a spiral galaxy with constant orbital speed $v(r) = v_0$ for $r > r_0$.

Step 1: Use circular motion

Step 2: Physical interpretation - Enclosed mass increases linearly with radius, even where visible matter is sparse. - This is evidence of a dark matter halo.

Step 3: Evaluate choices

(a) The mass density of the galaxy must decrease faster than $1/r^2$.
This is incorrect because if the density decreased faster than $1/r^2$, the orbital speed would decrease with radius. A flat rotation curve requires $\rho(r) \sim 1/r^2$ at large radii.

(b) The total enclosed mass becomes constant for $r > r_0$.
This is incorrect because $v^2 = GM(r)/r$ and $v =$ constant imply $M(r)\propto r$, so the enclosed mass continues to increase with radius.

(c) The luminosity profile directly traces the mass distribution.
This is incorrect because the flat rotation curve indicates most of the mass at large radii is dark matter, not luminous. Therefore, luminosity does not accurately trace total mass.

(d) The enclosed mass within radius $r$ grows approximately linearly with $r$.
This is CORRECT. From $v^2 = GM(r)/r$, constant $v$ implies $M(r)\propto r$, consistent with the presence of a dark matter halo.

(e) The galaxy cannot contain a central supermassive black hole.
This is incorrect because the flat rotation curve describes motion at large radii and does not constrain the mass distribution in the central region; a supermassive black hole can still exist at the center.

Answer: D