wiki/ simplerockets 1d

Simple Rocket Science

  1. Simple Rocket Science
    1. Getting Started
      1. The Simplest Case
        1. Rocket in free space
      2. Adding Gravity
        1. Burnout Velocity
        2. Burnout Height
    2. Ballistic Trajectory
    3. Bringing it all together
    4. References

Getting Started

Real spaceflight dynamics are complex. There are only a few basic forces involved, but some of them (namely air resistance) are non-conservative and tricky. In order to make a useful, accurate model of spaceflight you have to build a complex numerical approximation. However there is no need to be too complicated too quickly. A great place to start is to find an analytical solution for a rocket, solving for the height it will travel. Put everything in the 'up' axis and ignoring a ton of things will result in an easy place to start.

The Simplest Case

The true simplest case is a point-mass rocket that is in free space with no gravity, air-resistance and has a simplified model of propulsion. Describing this situation mathematically should result in the classic Tsiolkovsky rocket equation.

Rocket in free space

Consider a rocket in free space. There are no external forces. We will be nice and constrain the motion of the rocket in one direction. At some time the rocket starts to eject mass out of one end of the rocket. Through conservation of linear momentum this will propel the rocket forward.

Derivation

The motion of the rocket in our simple case will be determined by Newton's 2nd law. Usually written as F = ma since we have a variable mass system we use the more general form of

\vec{F} = \vec{\dot{p}}

We can safely ignore the vector part because all motion is in one dimension. Also we can re-write p-dot as dp/dt because we are going to use some arguments about infinitesimals to finish our derivation.

F = \frac{dp}{dt}

We can write an equation for dp from the diagram below because we know the momentums involved. From the definition of infinitesimal values

dp = p(t + dt) - p(t)

The momentum at some time plus dt will be the momentum of the rocket plus the momentum of the exhaust gas

2.2.png

In this diagram you can see we have a rocket of mass m moving at some velocity relative to an inertial reference frame v. At some time t + dt it will release an exhaust mass dm' at some velocity relative to the rocket ve. Many textbooks refer to ve as u. In a real rocket is an ideal "effective velocity" of the exit gasses and directly proportional to efficiency of the engine. It is related to the more common term Isp.

\mbox{mass of rocket at } t + dt = m - dm^\prime

\mbox{velocity of rocket at } t + dt = v + dv

\mbox{mass of exhaust mass} = dm^\prime

\mbox{velocity exhaust mass} = (v + dv) - v_e

Therefore

dp(t + dt) = (m - dm^\prime)(v + dv) + (dm^\prime)(v + dv - v_e)

p(t) = mv

dp = (m - dm^\prime)(v + dv) + (dm^\prime)(v + dv - v_e) - mv

Simplifying and throwing out higher order terms will reveal

dp = m dv - v_e dm^\prime

A positive exhaust mass is a mass lost from the rocket so we say that

dm^\prime = - dm

dp = m dv + v_e dm

In this simple case we have no external forces.

F dt = dp = 0

0 = m dv + v_e dm

Solving for dv gives

dv = - \frac{v_e}{m}dm

We can integrate both sides to find

\int^{v_f}_{v_0}dv = - v_e\int^{m_f}_{m_0}\frac{1}{m}dm

v_f - v_0 = - (v_e \ln (m_f) - v_e\ln (m_0))

Simplified this becomes the

Tsiolkovsky rocket equation

\Delta v = v_e \ln\left(\frac{m_0}{m_f}\right)

Since it is hard to look up values of ve in books about engines we can use its relation to Isp

v_e = g_0 I_{sp}

Where g0 is the acceleration due to gravity at the surface of the Earth and Isp is the specific impulse in seconds.

\Delta v = g_0 I_{sp} \ln\left(\frac{m_0}{m_f}\right)

Adding Gravity

Starting with our previous result of

Fdt = m dv + v_e dm

But now having an external force of gravity, mg

- mgdt = m dv + v_e dm

Solving for dv gives

dv = - \frac{v_e}{m}dm - g dt

Or as an equation of motion

\ddot x = - \frac{v_e}{m} \dot m - g

Integrating over v, t and m (Where tbo is the burn-time of the rocket)

\int^{v_{bo}}_{v_0}dv = - v_e\int^{m_f}_{m_0}\frac{1}{m}dm - g\int^{t_{bo}}_{0}dt

Gives an equation for the velocity at burnout

Burnout Velocity

v_{bo} = v_e \ln\left(\frac{m_0}{m_f}\right) - gt_{bo} + v_0

The term -gt is referred to as gravity loss. This represents the losses endured by launching in a gravity well. To maximize burnout velocity you want to minimize gravity loss, which means burning the fuel as fast as possible. This makes sense because when you spend a long time burning fuel you are wasting energy lifting unburnt fuel to a higher altitude rather than your payload.

In the real world usually rocket motors are categorized by thrust and not burn time. So it becomes useful to rewrite this equation in terms of a new parameter, thrust-to-weight ratio. We define thrust-to-weight ratio, Ψ, as the thrust (which we assume is constant) divided by the weight at liftoff, m0g.

\mbox{Thrust-to-Weight Ratio } \equiv \Psi = \frac{F_{thrust}}{m_0g}

If we realize that thrust is

F_{thrust} = v_e \dot m

then

\Psi = \frac{v_e \dot m}{m_0g}

We can find a relation to the time it takes to burn through the fuel. If we take the burn rate to be constant (again, not a bad assumption) then the time is to burn out is

\dot m = \frac{dm}{dt}

dt = \frac{dm}{\dot m}

\int^{t_{bo}}_{0} dt = \frac{1}{\dot m} \int^{m_f}_{m_0} dm

t_{bo} = \frac{m_f - m_0}{\dot m}

However we see that mf - m0 is negative! This okay because m-dot is secretly also negative. What we actually want is the fuel mass divided by a (positive) burn rate.

t_{bo} = \frac{m_0 - m_f}{\dot m}

We also want this in terms of mass ratio and thrust-to-weight ratio. So we do some clever math, first pulling out m0

t_{bo} = \frac{m_0}{\dot m} \left( 1 - \frac{m_f}{m_0} \right)

Now if we multiply by "one" we get

t_{bo} = \frac{v_e}{g} \frac{g m_0}{v_e \dot m} \left( 1 - \frac{m_f}{m_0} \right)

We recognize the term \frac{g m_0}{v_e \dot m} as the inverse of Ψ Thus we have the burnout time in reasonably terms

t_{bo} = \frac{v_e}{g} \frac{1}{\Psi} \left( 1 - \frac{m_f}{m_0} \right)

Plugging this in for t in our burnout velocity equation gets

v_{bo} = v_e \left[\ln\left(\frac{m_0}{m_f}\right) - \frac{1}{\Psi} \left( 1 - \frac{m_f}{m_0} \right)\right] + v_0

Now we can also introduce the symbol μ for the mass ratio (for brevity's sake) and replace ve with gIsp

v_{bo} = g I_{sp} \left[\ln(\mu) - \frac{1}{\Psi} \left( 1 - \frac{1}{\mu} \right)\right] + v_0

We see now that we have burnout velocity that is in general a function of three parameters: μ, Ψ and Isp

v_{bo} (\mu, \Psi, I_{sp})

Three variables and 3 dimensions is just begging for some graphing. Here is a graph showing the burnout velocity of a vertical rocket at an Isp of 250 [s]

Burnout Velocities for an Isp of 250. Click to enlarge

You can see that there a heavy dependence on mass ratio. But also to avoid gravity loss it is beneficial to have Ψ as high as you can too. A typical thrust-to-weight ratio is somewhere between 2 and 3 for large commercial rockets.

We can also talk about what effect changing Isp has on the burnout velocity. If we take the partial with respect to Isp we get

\frac{\partial v_{bo}}{\partial I_{sp}} = g \left[\ln(\mu) - \frac{1}{\Psi} \left( 1 - \frac{1}{\mu} \right)\right]

For a rocket with a μ of 10 and a Ψ of 2 we see that adding an extra second of Isp results in of 18.2 m/s of extra velocity.

\frac{\partial v_{bo}}{\partial I_{sp}} = 9.80665 [m/s^2] \left[\ln(10) - \frac{1}{2} \left( 1 - \frac{1}{10} \right)\right] = 18.2 [m/s/s]

Burnout Height

Now that we have done some work to find the burnout velocity we can find the burnout height much the same way. Starting with our differential equation for rocket motion we double integrate to find range

dv = - \frac{v_e}{m}dm - g dt

\iint^{h_{bo}}_{h_0} dv = - v_e \iint^{m_f}_{m_0}\frac{1}{m}dm - \iint^{t_{bo}}_{0}g dt

I had to look this one up in a table of integrals but the answer should be

h_{bo} = -v_e \left[ m_f \ln(m_f) - m_f - m_0 \ln(m_0) - m_0 \right] - \frac{1}{2} g t_{bo}^2 + h_0

This can be rearranged to look like

h_{bo} = v_e \frac{m_0}{\dot m} \left( 1 - \frac{m_f}{m_0} \left( \ln\left(\frac{m_0}{m_f}\right) + 1 \right)\right) - \frac{1}{2} g t_{bo}^2 + h_0

Doing the same substitution of t and using μ we get

h_{bo} = v_e \frac{m_0}{\dot m} \left( 1 - \frac{1}{\mu} ( \ln(\mu) + 1)\right) - \frac{1}{2} g \left( \frac{v_e}{g\Psi} \left( 1 - \frac{1}{\mu} \right)\right)^2 + h_0

h_{bo} = \frac{v_e^2}{\Psi g} \left( 1 - \frac{1}{\mu} ( \ln(\mu) + 1)\right) - \frac{v_e^2}{2g\Psi}\left(1 - \frac{1}{\mu}\right)^2 + h_0

Now making our Isp substitution

h_{bo} = \frac{g I_{sp}^2}{\Psi} \left[\left( 1 - \frac{\ln(\mu) + 1}{\mu}\right) - \frac{1}{2}\left(1 - \frac{1}{\mu}\right)^2\right] + h_0

Ballistic Trajectory

After burnout the rocket will continue to move upwards on a ballistic trajectory. Since we are considering gravity as a constant and ignoring air resistance we can use energy arguments to easily figure out the final height the rocket will go

KE_{bo} = PE_{final}

\frac{1}{2} m_f v_{bo}^2 = m_f g \Delta h

Solving for the change in height

\Delta h = \frac{v_{bo}^2}{2g}

h_f = \frac{v_{bo}^2}{2g} + h_{bo}

Bringing it all together

So if we really like math we can combine the ballistic and burnout equations to find the total height a rocket under constant gravity with no air resistance will travel (as a function of μ, Ψ and Isp)

h_f = g I_{sp}^2 \left[\frac{1}{2} \left[\ln(\mu) - \frac{1}{\Psi} \left( 1 - \frac{1}{\mu} \right)\right]^2 + \frac{1}{\Psi} \left[\left( 1 - \frac{\ln(\mu) + 1}{\mu}\right) - \frac{1}{2}\left(1 - \frac{1}{\mu}\right)^2\right]\right]

References

[1] Turner, Martin J.L. Rocket and Spacecraft Propulsion. Springer Praxis Books. [New York]: Praxis Publishing Ltd, Chichester, UK, 2005.

[2] Thomson, William Tyrrell. Introduction to Space Dynamics. New York: Wiley, 1961.