Generating plot of LV's performance as a function of the C3 parameter

Question

Following my previous question about Ariane 5 performance data for escape missions, I would like to know how the plot of launch vehicle capabilities like the next one are obtained:

More specifically, I would like to know how I can generate an approximation of these curves by using discrete data like the payload mass for a given C3.

My first idea was to use the rocket equation, but there is a lot of unknowns and the assumption that there are no external forces is clearly not respected for low altitudes.

My questions come down to:

• What model can I use to generate such a curve?
• What data are needed?

Answer 1

Despite @AdamWuerl's sage advice, I'll give it a go.

I'll do a single-stage, massless rocket body where mass is only propellant and payload as a zeroth-order spherical-cow attempt. If you needed better agreement, you'll have to model the staging and the boosters more faithfully.

$$v = v_{ex} \log(m_0/m)$$

inverts to

$$m = m_0 * \exp(-v/v_{ex}).$$

Let's say that $C_3$ is the excess $v^2$ above escape velocity from Earth:

$$C_3 = v^2 - v_{esc}^2,$$

$$v^2 = C_3 + v_{esc}^2,$$

so:

$$m = m_0 * \exp\left(-\sqrt{\frac{C_e + v_{esc}^2}{v_{ex}^2}}\right).$$

Your plot show the Delta IV(4450-14) has a payload mass of 4,500 kg at $C_3$=0. If you plug that convenient $C_3=0$ point into:

$$m_0 = m * \exp\left( \frac{v_{esc}}{v_{ex}} \right)$$

and use an exhaust velocity of 2.4 km/s (from Wikipedia), you get:

$$m_0 = 4,600 kg * \exp\left( \frac{11.2 km/s}{2.4 km/s} \right)$$

or about 489,000 kg, which is roughly right. The article puts various versions between 250,000 and 733,000 kg.

Putting $C_3$ = 25 km/s back into:

$$m = m_0 * \exp\left(-\sqrt{\frac{C_3 + v_{esc}^2}{v_{ex}^2}}\right).$$

gives $m$ = 2951 kg, which is roughly the 2500 kg shown in your plot.

So within the constraints of a single-stage, massless rocket body where mass is only propellant and payload, this is how you might do it. If you needed better agreement, you'll have to model the staging and the boosters more faithfully.

Answer 2

These plots are best made using a trajectory optimization program like POST or OTIS, which is used to model the LV to some appropriate level of fidelity. Jacks-or-better is engine performance, some simple aerodynamic tables, operational constraints like max dynamic pressure, angle-of-attack limits, and fairing separation conditions, and of course the mass properties of each stage to modeling staging events. Each point on the C3 curve is a separate run of this software targeting the specified escape energy. To achieve more velocity payload must be offloaded, eventually the point where there's no left, just a payload less upper stage.

Unfortunately these tools take quite a bit of expertise to run, and are also export controlled, but if you have a copy the necessary data is generally available or inferable.

Important factors that influence the magnitude and the shape of the curve are the overall size of the LV (the SLS line is higher than the Falcon 9 line), the number of stages (3-stage vehicles tend to h