Need advice on thin accretion disk model

Question

I'm a physics teacher and I'm building a simple accretion disk model to show to students in an astrophysics class (undergraduate level), as an exemple of physics modeling. I need to know if this model is credible enough, and what references there may be on that subject, at the undergraduate level (I didn't found anything usefull yet).

Consider a spherical body of mass $M$, standing at rest at the origin. A thin ring (disk) of total mass $m$ is rotating aound it, with internal radius $a$ and external radius $b > a$. I'm neglecting viscosity. The disk's surface density $\sigma$ is the following (this choice of function gives simple expressions for mechanical energy and angular momentum. See below) : \begin{equation}\tag{1} \sigma(r) = \frac{\alpha}{r^{\frac{3}{2}}}, \end{equation} where $a \le r \le b$. Note : I feel a bit unsecure with this arbitrary choice, so I need opinions on this. The disk's mass is thus : \begin{equation}\tag{2} m = \int_a^b \sigma(r) \, 2 \pi r \, dr = 4 \pi \alpha \, (\sqrt{b} - \sqrt{a}). \end{equation} This gives the constant $\alpha$, which will be usefull below : \begin{equation}\tag{3} \alpha = \frac{m}{4 \pi \, (\sqrt{b} - \sqrt{a})}. \end{equation} The mechanical energy of some particle in circular orbit of radius $r$ is simply this : \begin{equation} dE = dK + dU = -\, \frac{G M \, dm}{2 r}, \end{equation} so, the disk's total mechanical energy is easy to find : \begin{equation}\tag{4} E = \int dE = - \int_a^b \frac{G M}{2 \, r} \, \sigma(r) \, 2 \pi r \, dr = -\, \frac{G M m}{2 \, \sqrt{a \, b}}. \end{equation} The total angular momentum of the disk is this : \begin{equation}\tag{5} L = \int \sqrt{G M r} \, dm = \int_a^b \sqrt{G M r} \, \sigma(r) \, 2 \pi r \, dr = \frac{m}{2} \big( \sqrt{G M b} + \sqrt{G M a} \big). \end{equation}

Matter accretion : Now, I consider matter falling on the disk from the outside. I ask that the angular momentum (5) be conserved (energy (4) will not be conserved). At time $t = 0$, there's only a thin ring of internal and external radius $b$, of mass $m_0$. At time $t > 0$, the ring enlarge itself to a disk of internal radius $a(t)$ while the external radius $b$ stays the same. Mass $m$ is now a function of time : $m \Rightarrow m(t) \ge m_0$. Conservation of angular momentum (5) gives this constraint on the internal radius : \begin{equation}\tag{6} a(t) = \Big( \frac{2 m_0 - m(t)}{m(t)} \Big)^2 \, b. \end{equation} Notice that $a \rightarrow 0$ when $m \rightarrow 2 m_0$. This is puzzling me a bit. Why the factor of 2 ?

Finally, as a simple model, I consider a mass rising linearly with time : $m(t) = m_0 \, (1 + \lambda \, t)$. This gives the following internal radius for the accretion disk : \begin{equation}\tag{7} a(t)= \Big( \frac{1 - \lambda \, t}{1 + \lambda \, t} \Big)^2 \, b \le b. \end{equation}

While angular momentum of this model is conserved and there is no viscosity, energy isn't conserved since there is matter falling on the central body. The mass density (1) have been chosen since it gives simple analytical expressions (see equ. (2), (4), (5) and (6)).

Now is this model viable ? Is it "realistic" or at least convincing enough ? Any references for such kind of simple mechanical models ?

And how to physically justify the surface density (1), without resorting to the mathematical simplicity of the results ?

Answer 1

I think it is unwise to neglect the role of viscosity in any serious discussion of accretion disks. Viscosity is entirely responsible for the torques that transport angular momentum in the disk and allow matter to accrete, which is what makes them such efficient emitters of electromagnetic radiation, which is the main reason we care about them.

As a way to motivate the role of viscosity, start with an examination of gas rings in Keplerian orbits at various radii. You find that the angular velocity varies with radius, so each ring will shear with respect to its neighbors (contrast with rigid body rotation).

Rather than considering what happens with the addition of mass, it is first useful to consider how the transfer of angular momentum causes material at one radius to spread out over a range of radii, both larger and smaller than the initial one. If the Keplerian velocity structure is mostly preserved, the net result is a transfer of angular momentum to gas at larger radii in the disk, allowing some gas to fall toward the center. Here's a good opportunity to use some dimensional analysis. For kinematic viscosity $\nu$, and treating the spreading of the gas surface density as a diffusion process, on what timescale will gas be radially tranpsorted a distance comparable to the radius $R$ at which it started? Answer: $R^2 / \nu$

If you do want to consider what happens when you add mass, a good place to start is to determine the circular radius at which the material initially settles, which corresponds to the orbit with the specific angular momentum of the infalling material but with the least energy. The idea is that the energy will dissipate quickly, but the angular momentum will be conserved until the aforementioned viscous processes take over.

Beyond that, I think these lecture notes might be useful for introducing key concepts at an undergraduate level. Among other things, they help to derive the standard $r^{-3/4}$ law for the temperature of the disk at large radii, which is crucial for explaining their visible spectra.

As for the surface density, it turns out that the scaling is also $r^{-3/4}$ for the radii where the temperature is described by that law. Perhaps the most straightforward explanation for how to derive that is given in chapter of 5 of this textbook, which is certainly accessible to advanced undergraduates.