= Solution
The definition of the <virial radius of a dark-matter halo> immediately gives the <virial mass of a dark-matter halo>
$$
\boxed{M_v=\frac{4\pi}{3}v\rho_c^0r_v^3.}
$$
In an <Einstein-de Sitter universe>, spherical collapse gives $v=18\pi^2\simeq178$, conventionally rounded to about $200$.
For the <Navarro--Frenk--White profile>, put $x=r/r_s=cs$. Direct integration gives
$$
M(r)=4\pi\rho_c^0\delta_{\rm char}r_s^3
\left[\log(1+x)-\frac{x}{1+x}\right].
$$
Using $r_s=r_v/c$ and $\delta_{\rm char}=vc^3g(c)/3$ therefore yields
$$
\boxed{\frac{M(s)}{M_v}=g(c)
\left[\log(1+cs)-\frac{cs}{1+cs}\right].}
$$
At small radius the bracket is $(cs)^2/2+O((cs)^3)$, so $M(r)\propto r^2$, consistent with the central $\rho\propto r^{-1}$ cusp. At large radius it is $\log(cs)-1+o(1)$, so the mass diverges logarithmically unless the halo is truncated.
The primordial free streaming of <warm dark matter> erases small-scale density fluctuations and lowers the central phase-space density, tending to replace the smallest, earliest cold-dark-matter cusps by shallower central profiles or cores.
Since $V_v^2=GM_v/r_v$, the spherical potential that vanishes at infinity is
$$
\Phi(s)=-g(c)V_v^2\frac{\log(1+cs)}s.
$$
Indeed, $r\,d\Phi/dr=GM(r)/r$, and normalization at $s=1$ requires
$$
\boxed{g(c)=\left[\log(1+c)-\frac{c}{1+c}\right]^{-1}.}
$$
The <circular speed> is consequently
$$
\boxed{V^2(s)=V_v^2\frac{g(c)}s
\left[\log(1+cs)-\frac{cs}{1+cs}\right].}
$$
It rises as $s^{1/2}$ near the centre, peaks at $r\simeq2.16r_s$, and then declines approximately as $\sqrt{\log r/r}$. Increasing the <concentration of a dark-matter halo> moves the peak inward in units of $r_v$ and raises it relative to $V_v$. Thus an NFW curve can be fairly broad but is not exactly a <flat galaxy rotation curve>; stellar and gas contributions matter when comparing with an observed <galaxy rotation curve>.
Lower-mass haloes typically collapse earlier, when the cosmic background density is larger. Their characteristic inner densities are consequently larger relative to the present virial density, producing the <mass-concentration relation of dark-matter haloes> in which concentration decreases weakly with mass.
For a mass smaller by $1024=2^{10}$,
$$
M_v\simeq9.8\times10^8M_\odot,
\qquad
r_v=200\,1024^{-1/3}\ {\rm kpc}\simeq19.8\ {\rm kpc},
$$
and $c=10\,1024^{0.1}=20$. Hence
$$
\boxed{r_s=r_v/c\simeq0.99\ {\rm kpc}.}
$$
A halo of roughly $10^9M_\odot$ and kiloparsec scale naturally hosts a <dwarf galaxy>, possibly an extremely faint one if star formation is inefficient.
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