vehicle, such as cross-sectional area at the maximum diameter. Gravitation The gravitational force that a celestial body exerts on a space vehicle is modeled with the body and vehicle taken as point masses; the bodies (Earth, Moon, etc.) are simplified as spheres; and the mass of the vehicle is much smaller than the mass of the body so that its effect on the gravitational acceleration can be neglected. Therefore the gravitational force is calculated by: where: is the gravitational force (weight); is the space vehicle's mass; and is the radial distance of the vehicle to the planet's center; and is the radial distance from the planet's surface to its center; and is the gravitational acceleration at the surface of the planet g is the gravitational acceleration at altitude, which varies with the inverse square of the radial distance to the planet's center: Powered flight The equations of motion used to describe powered flight of a vehicle during launch can be as complex as six degrees of freedom for in-flight calculations, or as simple as two degrees of freedom for preliminary performance estimates. In-flight calculations will take perturbation factors into account such as the Earth's oblateness and non-uniform mass distribution; and gravitational forces of all nearby bodies, including the Moon, Sun, and other planets. Preliminary estimates can make some simplifying assumptions: a spherical, uniform planet; the vehicle can be represented as a point mass; solution of the flight path presents a two-body problem; and the local flight path lies in a single plane) with reasonably small loss of accuracy. The general case of a launch from Earth must take engine thrust, aerodynamic forces, and gravity into account. The acceleration equation can be reduced from vector to scalar form by resolving it into its tangential (speed ) and angular (flight path angle relative to local vertical) time rate-of-change components relative to the launch pad. The two equations thus become: where: F is the engine thrust; α is the angle of attack; m is the vehicle's mass; D is the vehicle's aerodynamic drag; L is its aerodynamic lift; r is the radial distance to the planet's center; and g is the gravitational acceleration at altitude. Mass decreases as propellant is consumed and rocket stages, engines or tanks are shed (if applicable). The planet-fixed values of v and θ at any time in the flight are then determined by numerical integration of the two