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# 1.2 Modelling protoplanetary discs

Material in a protoplanetary disc will be in orbit around a central star (or protostar). A first approximation to the motion of the material is that it is in so-called Keplerian motion, that is it obeys Kepler’s laws of planetary motion. In particular Kepler’s third law, which is a consequence of Newton’s law of gravity, states that the square of the orbital period is proportional to the cube of the orbital radius (assuming circular orbits, which is generally the case), i.e. P2 ∝ a3. As long as the orbiting particle has a mass that is small compared to that of the star, this may be expressed in the following equation:

where G is the gravitational constant (6.674 × 10-11 N m2 kg-2) and M* is the mass of the star. The Keplerian orbital speed is therefore the distance travelled in an orbit (the circumference of the orbit, 2πa) divided by the orbital period, P. This is therefore

Equation label:(Equation 1)

Figure 4 shows a schematic view of a protoplanetary disc, comprised mostly of molecular hydrogen. The rest of this section will aim to derive and solve the differential equations that govern the vertical (out-of-disc plane) gas-density profile as well as the radial dependence of the orbital (in-disc plane) velocity, which turns out to differ from the pure Keplerian motion. The following two subsections therefore will look in turn at how these two properties may be quantified.

Figure 4 Schematic illustration of a protoplanetary disc.
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