Equation Explained

• Burnout_Velocity = Final Velocity (Delta-vee) required by the missile to achieve it's range in km/sec.

• G_Rad: Solved independently below to simplify equation.

• Range_Angle: Solved independently below to simplify equation.

G_Rad: = (Gravity_Parameter / Burnout_Radius)

Gravity_Parameter: Gravitational Parameter of planet in km³/s².
Burnout_Radius: See Equation Below

Gravitational Parameters of Popular Objects (km3/s2)
Earth	398,600.4418
Moon	4,902.7779
Mars	42,828
Titan	8,978.2
Pluto	871
Ceres Asteroid	63.1

Range_Angle: = (Range_Surface / Planet_Radius)

Range_Surface: Missile range in kilometers.
Planet_Radius = Radius of planet in kilometers.

Burnout_Radius: = (Planet_Radius + Burnout_Altitude)

Planet_Radius = Radius of planet in kilometers.
Burnout_Altitude = Burnout altitude of missile in kilometers.

EXAMPLE: We want to calculate the ∆v required for a 7,000 km ICBM. The first step is to begin calculating the sub-variables, using a burn-out altitude of 330 kilometers. For BurnoutRadius, since we know Earth's radius is 6,378 km; BurnoutRadius solves for: BurnoutRadius: = (6378 + 330) = 6,708 For Grad, since we know Earth's Gravitational Parameter is 398,600, and because we solved for BurnoutRadius earlier; Grad solves for: GRad: = (398,600 / 6,708) = 59.42 For RangeAngle, since we know Earth's radius is 6,378 km, RangeAngle solves for RangeAngle: = (7,000 / 6,378) = 1.097 Now that we have solved the sub-variables; we can solve the main equation: Thus, our hypothetical 7,000 km ICBM would require a ∆v of 6.05 km/sec². However, it is a good rule of thumb to add an extra 0.5 km/sec² to your estimates to take into account air drag during the in-atmosphere phase, and to provide a margin for under-performing components.

Bibliography:
Space Vehicle Design by Griffin, referencing Fundamentals of Astrodynamics by Bate, Mueller and White