Rocket Equation Explorer

Neil deGrasse Tyson recently discussed the Rocket Equation on Star Talk, but he didn't quite get the details right. Let's check his claim that "the amount of fuel you need to deliver a certain payload grows exponentially for every extra pound of payload"

The Tsiolkovsky rocket equation relates a rocket’s change in velocity to its engine exhaust velocity and the ratio between its starting and final mass:

Δv = v_e ln(m₀ / m_f)

Δv is the required change in velocity, v_e is the exhaust velocity, m₀ is the initial mass, and m_f is the final mass.

The initial mass m₀ includes all components of the rocket: fuel, structure, and payload. That is:

m₀ = m_fuel + m_structure + m_payload

For simplicity, we combine the structure and payload into a single term m_dry:

m₀ = m_fuel + m_dry

After all fuel is burned, only the dry mass remains, so:

m_f = m_dry

Substituting these into the rocket equation and solving for fuel mass gives:

m_fuel = m_dry(e^{Δv / v_e} − 1)

Use the two scenarios below to see how changing the final mass, required Δv, or exhaust velocity changes the amount of fuel required.

Scenario A

Δv required:

km/s

Final mass (m_dry):

tons

Exhaust velocity:

km/s

Fuel required

- tons

Scenario B

Δv required:

km/s

Final mass (m_dry):

tons

Exhaust velocity:

km/s

Fuel required

- tons

Fuel Required vs Final Mass

Here is a plot of fuel mass vs dry mass from the Rocket Equation. Contrary to Dr. Tyson's claim and description, it is a linear relationship. An easy way to think of this is to imagine that if you want to double the payload of a rocket you could just send a second rocket. Double the payload, double the fuel. Linear.

Fuel Required vs Δv

The relationship that is exponential is between delta-v and fuel. In other words, it doesn't take exponentially more fuel to carry more mass but rather to take that same mass farther/faster.