We consider a disk around a star of mass . At a distance from that star is a planet of mass . The planet interacts gravitationally with a fluid element at a radius such that . Both the fluid element and the planet are moving along circular Keplerian orbits around the star, so the velocity difference between them is

Most of the deflection occurs when the distance between the planet and the fluid element is about , and the time interval when both are that close is about . The velocity deflection perpendicular to the fluid element's original direction of motion is therefore

Conservation of energy dictates that there must also be a change in the component of velocity parallel to the original trajectory

If the mass of the fluid element is , then the exchange of angular momentum is . The mass of the fluid element is given by , where is the width of the annulus. The time it takes the annulus to interact with the planet is just

The torque between the annulus and the planet is given by

The total torque is

The lower limit on the integration is the minimum of either the scale height of the disk or the radius of the Hill sphere. In the case of the latter,

and