Shakura-Sunyaev Accretion Disc Temperature

What you are seeing: the temperature profile of a steady, optically thick, geometrically thin accretion disc around a compact object. The Shakura-Sunyaev (1973) closed form is

The bracket factor vanishes at the inner edge (so $T(R_\text{in}) = 0$ even though the bare $r^{-3/4}$ scaling would diverge from below); the maximum of $T(r)$ occurs at $r = (49/36) R_\text{in}$ .

The top panel shows $T(r)$ in cyan and the bare $r^{-3/4}$ scaling in dashed orange. The bottom panel is a face-on rendering of the disc with annulus colors set by local temperature. The hot inner ring shows the temperature peak and the gradual cooling toward larger radii.

Figure 1. Shakura-Sunyaev temperature profile. Method: closed-form bracketed power law with bare

r^{-3/4}

overlay.

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WHAT TO TRY

Watch the T ~ r^-3/4 Shakura-Sunyaev profile: the inner disk is hottest and bluest, falling off outward. Each annulus is colored by its local blackbody temperature.
Extend r_out: the cool outer disk adds red flux without touching the hot inner peak, which is why the integrated spectrum is a broadened multi-temperature blackbody.
Switch the view to set the radial temperature curve against the face-on disk image: the same r^-3/4 law drives both.