Missile Aerodynamics

This article describes the aerodynamics of missiles in detail.

Gravity
Missiles are unaffected by gravity as long as they are under power. We will ignore gravity for all derivations here.

Mass
Missiles have a mass of 0.1 t per module, or 0.2 t per block. Thumper Warheads weigh an additional 0.2 t. This would make missile thrust displayed in kN fps. Let $$\ell$$ be the length of the missile in m, plus 1 if the missile has a Thumper Warhead.

$$m = 0.2 \ell$$

Thrust
In this article, we will use kN for thrust $$T$$. This is 1/40 the displayed thrust $$T_D$$:

$$T = \frac{T_D}{20}$$

The displayed thrust $$T_D$$ is 1000 per Short-Range Thruster plus a variable amount from Variable Thrusters in air, or 250 per Torpedo Propeller in water. The acceleration due to thrust is

$$a_T = \frac{T}{m} = \frac{T}{0.2 \ell} = \frac{T_D}{8 \ell}$$

Specific impulse
For air thrusters, one Fuel Tank carries 125 kN s of fuel and weighs 0.1 t. This gives a specific impulse of about 127 s.

Endurance
The endurance $$t_e$$ is the time that the missile is powered. This is equal to the fuel $$F$$ (5000 per tank) divided by the fuel consumption (per second), but no longer than the missile's lifetime. The fuel consumption is equal to the total displayed thrust from air thrusters, plus 25 per Torpedo Propeller.

Lifetime
The lifetime $$t_\ell$$ is the maximum time the missile can exist. This is 60 s plus 180 s per Regulator.

Fluid density $$\rho$$
Important for drag. This is normalized to air density at sea level and is linearly interpolated between the following values:

Drag
Apart from thrust, drag is the major force acting on a missile.

Drag coefficient
Let $$c_{di}$$ be the base drag of the $$i$$th component, starting from 1 at the nose. The drag coefficient is then

$$c_d = \frac{ 1 + 10 \sum_i \frac{c_{di}}{i} } {1000}$$

Displayed drag
The displayed drag is the rate of velocity loss as a proportion of velocity.

$$\max \left( \mu, \rho v c_d \right)$$

where $$\mu = 0.1$$ represents linear drag, while $$\rho v c_d$$ represents quadratic drag.

Linear and quadratic drag
We can get the drag force by multiplying by $$-v$$ to get the drag acceleration, and then by the mass of the missile $$m = 0.2 \ell$$ to get the drag force.

At low speeds, the drag is proportional to velocity. Linear drag is


 * Acceleration: $$a_{d1} = -\mu v = -0.1 v$$
 * Force: $$F_{d1} = -\mu mv = -0.02 \ell v$$

regardless of whether the missile is in air or water. At high speeds, the drag is proportional to the square of velocity. Quadratic drag is


 * Acceleration: $$a_{d2} = -\rho v^2 c_d$$
 * Force: $$F_{d2} = -m \rho v^2 c_d = - 0.2 \ell \rho v^2 c_d$$

The actual drag acceleration and force is the greater of the linear drag and the quadratic drag.

Initial speed
The initial speed $$v_0$$ is determined by the following:
 * The Launch Pad itself gives a (usually negligible) 0.75 kN·s impulse. This translates into 3.75 m/s divided by the length of the missile in blocks.
 * 35 m/s per Ejector Add On attached to the Launch Pad.
 * 15 m/s per Short-Range or Variable Thruster. This is not affected by the Variable Thruster's settings.
 * 50 m/s per Torpedo Propeller.

Terminal speed
A missile is at terminal speed when the drag force is equal to the thrust $$T$$. We can compute a terminal speed for both linear and quadratic drag.

Linear terminal speed is:

$$v_{z1} = \frac{T}{\mu m} = \frac{T_D}{0.8 \ell}$$

Quadratic terminal speed is:

$$v_{z2} = \sqrt{\frac{T}{m \rho c_d }} = \sqrt{\frac{T_D}{8 \ell \rho c_d }} $$

The actual terminal speed is the lesser of the two. Above this speed the missile will slow down, and below this speed it will speed up.

Critical speed
The speed at which drag switches from linear to quadratic is the critical speed. This is

$$v_c = \frac{\mu}{\rho c_d}$$

The critical thrust, which makes $$v_z = v_c$$, is

$$T_c = \mu m v_c$$

The critical displayed thrust is

$$T_{Dc} = 40 T_c = 40 \mu m v_c = \frac{0.12 \ell}{\rho c_d}$$

This speed is important for range calculation and optimization. At up to the critical speed, range is roughly independent of speed; above it, range decreases. Therefore, the critical thrust is the minimum recommended thrust---use higher if necessary, but not lower.

Turn speed
The turn speed displayed in-flight is equal to

$$\frac{180}{\pi} \omega = 10 \sqrt{v} \frac{f}{\ell}$$

where $$f$$ is the number of fins. This is a measure of angular velocity and is in units of degrees per second. The turn radius of a missile is then

$$r = \frac{v}{\omega} = \frac{18}{\pi} \sqrt{v} \frac{\ell}{f} \approx 5.73 \sqrt{v} \frac{\ell}{f} $$

Simple estimate
Range can be estimated by simply multiplying the terminal speed $$v_T$$ by the burn time $$t_e$$:

$$R \approx v_t t_e$$

This tends to be an overestimate of the range, since typically the missile's initial speed will be less than the terminal speed.

Exact computation
The exact range (for a constant altitude) can be computed by considering piecewise segments of the flight below and above the critical speed.

Linear
Below the critical speed (linear drag), the missile's speed decays exponentially towards the linear terminal speed.

$$v \left( t \right) = v_0 + \left( v_0 - v_{z1} \right) e^{-\mu t}$$

To get the distance as a function of time, we integrate:

$$x \left( t \right) = v_0 t + \frac{ v_0 - v_{z1} }{\mu} \left( e^{-\mu t} - 1\right)$$

Quadratic, below terminal speed
Above the critical speed (quadratic drag) but below the quadratic terminal speed, the missile's speed decays towards the quadratic terminal speed as a hyperbolic tangent.

Let the convergence rate be

$$k = \sqrt{a_T \rho c_d} $$

Then

$$v \left( t \right) = v_{z2} \tanh \left(k \left( t + t_0 \right) \right) $$

where $$t_0$$ is set so that $$v \left( 0 \right)$$ is equal to the initial velocity.

Again we integrate to find the distance as a function of time:

$$x \left( t \right) = \frac{1}{\rho c_d} \log \left( \cosh \left(k \left( t + t_0 \right) \right) - \cosh \left(kt_0\right) \right)$$

Quadratic, above terminal speed
The only difference is that the hyperbolic functions are swapped:

$$v \left( t \right) = v_{z2} \coth \left(k \left( t + t_0 \right) \right) $$

$$x \left( t \right) = \frac{1}{\rho c_d} \log \left( \sinh \left(k \left( t + t_0 \right) \right) - \sinh \left(kt_0\right) \right)$$