= Solution
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>, $\Omega_K=(GM_\star/r^3)^{1/2}$. 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 gives
$$
\frac{\partial P_g}{\partial z}=-\rho_g\Omega_K^2z.
$$
For a locally vertically isothermal <ideal gas>, $P_g=\rho_gc_s^2$ with $c_s^2=k_BT/(\mu m_H)$, where $\mu$ is <mean molecular weight>. Integration gives
$$
\boxed{\rho_g(r,z)=\frac{\Sigma_g(r)}{\sqrt{2\pi}H}
\exp\!\left(-\frac{z^2}{2H^2}\right),\qquad H=\frac{c_s}{\Omega_K}.}
$$
Here $\Sigma_g$ 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.
\Image[/past-exam-of-the-exoplanets-and-planetary-systems-course-of-the-university-of-cambridge/2016/paper-1-disk-structure.png]
{title=Gas, settled solids and condensation fronts in an original schematic <protoplanetary disk>}
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.<Lecture notes on the formation and early evolution of planetary systems by Philip J. Armitage>[2007 v1, Section II.D, condensation sequence; Section III.A, settling and the fiducial global gas-to-solid ratio]{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 $v_z\simeq-\Omega_K^2t_sz$. 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.<Lecture notes on the formation and early evolution of planetary systems by Philip J. Armitage>[2007 v1, Section II.C.6, disk dispersal and accretion/photoevaporation timescale comparison]{ref}
\b[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.
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