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A protoplanetary disk is a flattened, rotating reservoir of gas and solids around a young star. Its radial gravity is usually dominated by the host, making its rotation close to Keplerian rotation, ${\Omega }_K=(G{M}_{\star }/r^3{)}^{1/2}$. Gas pressure provides vertical support and a small radial correction to the orbital speed; it does not make the disk a nonrotating spherical atmosphere.

At height $|z|\ll r$, vertical hydrostatic equilibrium in the stellar gravitational field givesFor a locally vertically isothermal ideal gas, $P_g={\rho }_g{c}_s^2$ with $c_s^2=k_{B}T/(\mu m_H)$, where $\mu$ is mean molecular weight. Integration givesHere ${\Sigma }_g$ is the vertically integrated gas surface density of a disk, and $H$ is the isothermal disk scale height. The disk aspect ratio $H/r=c_s/v_K$ is small in a geometrically thin disk. For $T\propto r^{-q}$ it scales as $r^{(1-q)/2}$, so $q<1$ produces disk flaring. This illustrative local vertical model does not determine a unique radial surface-density law or temperature normalization from the information given.

The gas is predominantly molecular hydrogen and helium. A much smaller component consists of heavier elements, partitioned between vapor, solid refractory grains and condensed volatile ices. A gas-to-solid mass ratio of order $100$ is a common initial fiducial model, not a universal measurement applying at every disk radius or evolutionary stage. Solids can grow and settle while gas accretes or disperses, changing the local ratio substantially.^[ref]

Thermal structure organizes composition. In the hottest inner region, grains can sublimate and much of their material remains in the gas. Refractory silicate minerals and metallic grains survive at lower temperatures; farther out, water and then other sufficiently volatile species can condense when their local condensation conditions are met. The water snow line can raise the solid inventory available to planet formation. Its location depends on stellar irradiation, accretion heating, opacity and gas pressure, and evolves with time; it is not fixed merely by naming a star or adopting a solar-system distance.

Stellar radiation heats the exposed surface of a passive disk. Beneath it, shielding can leave a cooler, molecule-rich interior with a higher mass density. In the inner actively accreting disk, dissipation can instead make the optically thick midplane the hottest region. These are distinct heating regimes; a vertically isothermal profile is an analytic approximation rather than a claim that every real layer has the same temperature. Dust opacity controls how radiation is absorbed and re-emitted. Infrared and millimetre emission probe solid material, while spectral lines probe gas; neither observable alone supplies a model-free inventory of the whole disk.

Dust settling in an astrophysical disk makes larger, less tightly coupled particles more concentrated toward the midplane, while turbulence and mixing oppose that settling. For an aerodynamic stopping time $t_s$ short compared with ${\Omega }_K^{-1}$, terminal vertical balance gives $v_z\simeq -{\Omega }_K^2{t}_{s}z$. Small grains remain better coupled to the gas. Radial drift also changes the distribution of solids and their access to snow lines and embryos.

The disk evolves through accretion and angular momentum transport, potentially involving turbulent stresses, magnetorotational instability or magnetically driven winds where conditions permit. Protoplanetary disk dispersal eventually removes gas through accretion and mass loss, including photoevaporation. In standard evolutionary pictures this occurs on a scale of a few million years, with substantial system-to-system variation. That finite reservoir lifetime is central to whether an embryo acquires a massive gaseous envelope.^[ref]

The essential structure is a nearly Keplerian gas disk with vertical pressure support, radial and vertical thermal gradients, and an evolving solid population concentrated by settling, growth and drift. Its composition is set jointly by the original elemental inventory, local condensation conditions and transport.

The circular Keplerian orbit has speed $v_K=(GM/r{)}^{1/2}$. Equating it with the heated gas sound speed givesThis is the photoevaporative gravitational radius, where thermal and orbital binding energies have the same order of magnitude. A circular orbit has specific mechanical energy $-GM/(2r)$. Heating adds thermal energy and, in a fluid outflow, available specific enthalpy of order $c^2$. For example, if $c$ is the adiabatic sound speed, an ordinary ideal gas has enthalpy $c^2/(\gamma -1)$; for $\gamma =5/3$ this is $3c^2/2$. At $r\ge r_g$, this more than compensates the circular-orbit binding energy. Equivalently the hot hydrostatic scale height satisfies $H_{\mathrm{hot}}/r\sim c/v_K\gtrsim 1$, so a thin bound surface layer cannot be maintained. With continued irradiation, the gas can expand into a thermal wind: photoevaporation removes disk material.

The condition is a thermal binding scale, rather than an assertion that the sound speed equals the ballistic escape speed, which is $\sqrt{2}{v}_K$. Detailed wind launching can change the numerical critical radius by factors of order unity. Using exactly the supplied numerical estimates, $c=10^6\mathrm{cm}{\mathrm{s}}^{-1}$, and therefore

The wind removes $2\pi rW(r)dr$ of mass per unit time from an annulus; $W$ is already the surface-density loss term in the supplied mass conservation equation, so there is no additional two-face factor. Integrating the photoevaporation profile givesThe convergence at infinity is important: the loss is concentrated near the photoevaporative gravitational radius. Using $r_g=1.4\times 10^{14}\mathrm{cm}$ yields ${\dot{M}}_w\simeq 2.46\times 10^{17}\mathrm{g}{\mathrm{s}}^{-1}$, or about $3.9\times 10^{-9}{M}_{\odot }\mathrm{y}{\mathrm{r}}^{-1}$. The initial disk mass is $2\times 10^{30}\mathrm{g}$, so the wind-only depletion time isThis estimate treats the heated area and wind normalization as fixed and neglects additional removal through stellar accretion. Once the disk shrinks, its wind rate and geometry need not remain constant.

When a protoplanetary disk's inward viscous supply approaches its wind-loss rate, photoevaporation can prevent replenishment of the inner disk. That region drains on a viscous timescale, leaving a gap or hole. A negative density in a formal steady state calculation signals the breakdown of that approximation, not a physical continuation. An exposed outer edge can subsequently disperse rapidly.

The thermal binding scale defined by equating heated gas sound speed to circular Keplerian orbit speed. It is $r_g=GM/c_s^2$. The circular-orbit binding energy is $GM/(2r)$ per unit mass, while the gas specific enthalpy is of order $c_s^2$. A heated disk surface at radii comparable to or larger than $r_g$ can therefore drive photoevaporation. The actual launching radius can differ by a factor of order unity; $c_s$ is not the ballistic escape speed.

Removal of disk gas by accretion onto the host star, winds and photoevaporation. The finite availability of gas limits which solid cores can acquire massive envelopes. The disk's evolutionary state matters more than a single universal dispersal time.