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# 1.4 Radial dependence of the orbital velocity

Having considered the vertical density profile of the disc, we now turn to the radial dependence of the orbital velocity . In the radial direction, in addition to the gravitational force, there is also a force due to the pressure gradient. Hence, the net centripetal acceleration of a small volume of gas in the disc on an assumed circular orbit at radius r is

Here, since M* is much greater than the total mass of the disc inside the orbital radius. Therefore, the orbital speed vorb of the gas in the disc has two components: one due to the Keplerian speed , and one due to this extra pressure gradient. It is given by:

Equation label:(Equation 9)

Usually, in the disc, so the gas will behave as if it was feeling a slightly lower gravitational pull from the star, and its orbital speed will be sub-Keplerian, that is, . The following activity shows how to quantify the magnitude of the deviation between the actual orbital speed of the gas and its Keplerian speed.

## Activity 1

Using Equation 9 and approximating the pressure gradient as , where n is a dimensionless constant:

1. Write an approximate expression for as a function of the radius, scale height and Keplerian speed, .

2. Calculate the difference between the orbital and Keplerian speeds, , at a radius of 1 au from a star of the same mass as the Sun, for a disc of constant aspect ratio H/r = 0.05 and n = 3. (You may assume 1 au = 1.496 × 1011 m, 1 M☉ = 1.99 × 1030 kg, and G = 6.674 × 10-11 N m2 kg-2.)

### Answer

1. From Equation 3, the pressure of the gas Pgas is linked to the sound speed cs such that . Using this, the expression for the pressure gradient becomes:

   then substituting this into Equation 9 gives

   Finally, using the fact that

   the requested expression is obtained as:

   Equation label:(Equation 10)

2. From Equation 10, the difference in velocities is

   So the difference is only about 0.4% of the Keplerian velocity.

   To evaluate Δv at r = 1 au we need to calculate the Keplerian velocity vK at 1 au:

   which gives . So the difference between the orbital and Keplerian speeds at this radius is about 100 m s-1.

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