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A classically accelerating Coulomb electron radiates and loses orbital energy, so a stable nonradiating orbit is impossible within that unmodified theory. Bohr instead postulates stationary nonradiating orbits, quantized angular momentum $\mu vr=n\hslash$ and photon emission/absorption with $h\nu =E_i-E_f$. Here $\mu =m_{e}M/(m_e+M)$ describes relative motion. The angular-momentum condition is the standing-wave relation $\boxed{2\pi r=n{\lambda }_{\mathrm{dB}}}$ for the relative orbit. These are additional quantum postulates, not consequences of classical radiation theory.

For a bound Kepler orbit, specific orbital energy $v^2/2-\mu /r$ is negative. Therefore $\mu >v^{2}r/2$. A projected separation $R_{\perp }\le r$ and a systemic-motion-corrected line-of-sight speed $|v_{\mathrm{los}}|\le v$ give the displayed lower bound. One position and one spectral velocity already constrain the mass in this model; a fitted orbit gives much more. The bound assumes a bound orbit in a point-mass potential; it is not an unrestricted enclosed-mass formula for an arbitrary extended potential.

An impulse directed along the position vector relative to a specified origin. Its angular impulse vanishes because $\Delta \mathbf{J}=\mathbf{r}\times \Delta \mathbf{p}=0$. It can nevertheless change kinetic energy; if an initially circular Kepler orbit receives radial velocity $w$, the added specific orbital energy is $w^2/2$.

A radial impulse that multiplies the initial circular kinetic energy by $\beta >1$ adds radial velocity $\pm \sqrt{GM/r_0}\sqrt{\beta -1}$ and leaves specific angular momentum $j^2=GM{r}_0$. The new specific orbital energy is $GM(\beta -2)/(2r_0)$, yielding orbital eccentricity $e=\sqrt{\beta -1}$ and semilatus rectum $p=r_0$. For $1<\beta <2$, the apsidal radii are $r_0/(1\pm e)$; for $\beta \ge 2$ there is no finite apocentre. A nonzero radial kick occurs away from an apsis, so the old circular radius is not automatically the new periapsis.

For the steady zero-inner-torque disk, the standard thin-disk dissipation flux already includes both faces. Integrate it over projected annuli, rather than multiplying by a second factor of two:This is the decrease in orbital energy per unit time as matter arrives from arbitrarily large radius on a circular orbit. Half the decrease in gravitational potential energy remains as orbital kinetic energy at the inner edge. Accretion onto a slowly rotating stellar surface can release additional energy in a boundary layer; a black-hole inner boundary requires the appropriate relativistic treatment.^[ref]

For a finite outer radius $R_{\mathrm{out}}$, integration of this same specified flux gives insteadIt tends to zero as $R_{\mathrm{out}}\downarrow R_{\mathrm{in}}$ and to $L_{\mathrm{thin}}$ as $R_{\mathrm{out}}\to \infty$. This finite-annulus result inherits the same steady torque profile; it does not impose an additional zero-torque outer boundary.