Planar reentry equations

The planar reentry equations are the equations of motion governing the unpowered reentry of a spacecraft, based on the assumptions of planar motion and constant mass, in an Earth-fixed reference frame.

Definition

The equations are given by:where the quantities in these equations are:

• is the velocity
• is the flight path angle
• is the altitude
• is the atmospheric density
• is the ballistic coefficient
• is the gravitational acceleration
• is the radius from the center of a planet with equatorial radius
• is the lift-to-drag ratio
• is the bank angle of the spacecraft.

Simplifications

Allen-Eggers solution

Harry Allen and Alfred Eggers, based on their studies of ICBM trajectories, were able to derive an analytical expression for the velocity as a function of altitude. They made several assumptions:

1. The spacecraft's entry was purely ballistic .
2. The effect of gravity is small compared to drag, and can be ignored.
3. The flight path angle and ballistic coefficient are constant.
4. An exponential atmosphere, where , with being the density at the planet's surface and being the scale height.

These assumptions are valid for hypersonic speeds, where the Mach number is greater than 5. Then the planar reentry equations for the spacecraft are:

Rearranging terms and integrating from the atmospheric interface conditions at the start of reentry leads to the expression:

The term is small and may be neglected, leading to the velocity:

Allen and Eggers were also able to calculate the deceleration along the trajectory, in terms of the number of g's experienced , where is the gravitational acceleration at the planet's surface. The altitude and velocity at maximum deceleration are:

It is also possible to compute the maximum stagnation point convective heating with the Allen-Eggers solution and a heat transfer correlation; the Sutton-Graves correlation is commonly chosen. The heat rate at the stagnation point, with units of Watts per square meter, is assumed to have the form:

where is the effective nose radius. The constant for Earth. Then the altitude and value of peak convective heating may be found:

Equilibrium glide condition

Another commonly encountered simplification is a lifting entry with a shallow, slowly-varying, flight path angle. The velocity as a function of altitude can be derived from two assumptions:

1. The flight path angle is shallow, meaning that: .
2. The flight path angle changes very slowly, such that .

From these two assumptions, we may infer from the second equation of motion that:

See also

• Atmospheric entry
• Hypersonic flight

References

Further reading

• Regan, F.J.; Anandakrishnan, S.M. (1993). Dynamics of Atmospheric Re-Entry. AIAA Education Series. pp. 180-184.