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A lowered Maxwellian velocity distribution subtracts the value of a Maxwell-Boltzmann velocity distribution at the escape velocity and sets the result to zero beyond escape. This gives a nonnegative phase-space distribution function vanishing continuously at the boundary. If $A=n_0/(2\pi {\sigma }^2{)}^{3/2}$ and $x_e=v_e/(\sqrt{2}\sigma )$, its actual number density is $n=n_0[\mathrm{erf}{x}_e-2x_e{e}^{-x_e^2}/\sqrt{\pi }-4x_e^3{e}^{-x_e^2}/(3\sqrt{\pi })]$. Thus $n_0$ is a normalization parameter, not generally the integrated density. Integrating only speeds below $V<v_e$ gives the same bracket with the first two terms at $X=V/(\sqrt{2}\sigma )$ and the last term $-4X^3{e}^{-x_e^2}/(3\sqrt{\pi })$.

Take $V=|\mathbf{V}|$ to be the test body's instantaneous speed relative to the nonstreaming background, rather than an ensemble root mean square speed. Use a local isotropic galactic distribution function $f(v)$ normalized as a number distribution, and assume independent encounters and a large, approximately constant Coulomb logarithm in stellar dynamics.

For a background particle with relative incoming velocity $\mathbf{w}=\mathbf{V}-\mathbf{v}$, a gravitational scattering angle $\chi$ obeysIn the centre-of-momentum frame, the test body's share of the relative velocity is $m/(M+m)$. The azimuthal average cancels transverse changes, leaving a change parallel to the incoming relative velocity,The number of encounters per unit time in $d^{3}v$ and impact interval $db$ is $f(v)d^{3}vw2\pi bdb$. Integrating the longitudinal change from $b=0$ to $b_m$ givesAt leading logarithmic order replace the logarithm by $2\ln \Lambda$, evaluated at a representative speed. For isotropic $f$, the velocity-shell identity for dynamical friction isOne proof takes the divergence with respect to $\mathbf{V}$: the inverse-square vector field has divergence $4\pi {\delta }^{(3)}(\mathbf{V}-\mathbf{v})$, whose velocity integral is $4\pi f(V)$, and rotational symmetry plus the divergence theorem determines the radial field. Exterior velocity shells give zero field. Consequently the systematic Chandrasekhar dynamical friction is antiparallel to motion, withIn the massive-test-body approximation $M\gg m$, this becomesThe finite-test-mass dynamical friction coefficient shows the needed qualification: $M>m$ alone does not make $M+m=M$. The quoted expression also neglects diffusion and the velocity dependence of the logarithmic cutoff. For the lowered Maxwellian velocity distribution, set $f=0$ for $v>v_e$; continuing the printed subtraction beyond escape would give a negative distribution. If $V>v_e$, the integral stops at $v_e$.

Inside the uniform sphere, $V_c^2=3{\sigma }^2{r}^2/r_0^2$. Thus, with $X=V_c/(\sqrt{2}\sigma )$, part (b) givesThe Gaussian integral needed for the lowered Maxwellian velocity distribution isSubstitution $v=\sqrt{2}\sigma y$ therefore yieldsSince $V_c<v_e$ throughout the core, the integrand is positive and so is $I(X)$ for $X>0$. Using $m{n}_0={\rho }_0$, part (a)'s massive-test-body drag magnitude isThe slow drift of a circular orbit is determined by torque, not by equating its changing circular speed to the tangential drag. Here $V_c=\omega r$, with constant $\omega =\sqrt{4\pi G{\rho }_0/3}$. The specific angular momentum $j=r{V}_c=\omega r^2$ obeysHence the positive local decay timescale isThere is a genuine factor-of-two discrepancy with the requested denominator $32{\pi }^2$. That denominator would give $V_c/(2D)$, which is the timescale for a flat galaxy rotation curve, not this uniform-density core. It also follows from the incorrect replacement ${\dot{V}}_c=-D$, which drops the $V_c\dot{r}$ term in $d(r{V}_c)/dt$. The literal signed definition $r/(2\dot{r})$ is negative for an inward drift; it equals $-{\tau }_{+}$. A positive decay time requires the absolute value or a minus sign.

This is the slow circular inspiral under tangential drag calculation. Its assumptions require $D$ small enough for nearly circular adjustment and a meaningful large Coulomb logarithm in stellar dynamics; the leading-log approximation is not uniform at $V_c\to 0$. Also, integrating the truncated distribution all the way to $v_e$ does not give exactly $n_0$. The model uses $n_0$ as the stated core normalization; exact self-consistency of a sharply truncated uniform core with that distribution is not implied.