Computing Delta-V required
for Ballistic Missiles

(Created October 2009)
(Last Updated July 2016)

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

Pre-Calculated Delta V requirements for Ballistic Missiles on Earth

Notes: These were calculated with the equations below.

Missile

Range /Apogee (km)

Delta-V required

Atlas F ICBM
(range/apogee from Standard Missile Characteristics)

12,570 km range
1,413 km apogee

5,680 m/sec

Thor IRBM
(public sources)

2,400 km range
450 km apogee

4,257 m/sec

Iraqi Al Hussein
(public sources)

644 km range
151 km apogee

2,420 m/sec

Equation to Compute Delta-V for a Given Range

Equation Explained

GRad: = (GravityParameter / BurnoutRadius)

GravityParameter: Gravitational Parameter of planet in km3/s2.
BurnoutRadius: 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

RangeAngle: = (RangeSurface / PlanetRadius)

RangeSurface: Missile range in kilometers.
PlanetRadius = Radius of planet in kilometers.

BurnoutRadius: = (PlanetRadius + BurnoutAltitude)

PlanetRadius = Radius of planet in kilometers.
BurnoutAltitude = 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.