Equations

${\displaystyle }$This is the master list for equations!

Δ v

$\Delta v=I_{sp}\times g_{0}\times \ln {({\frac {M_{t}}{M_{d}}})}$

Where:

• M _t is total mass.
• M _d is total mass minus fuel mass.
• "In" is the natural logarithm.

See the Δv page for a comprehensive explanation of Δv.

Rocket Efficiency 

According to Stef:

Top to Bottom

$E_{r}=M+2M^{n}$

Where

• E_r is Efficiency of the rocket, and

• M_n < is Mass in tonnes of the succeeding stages.

Bottom to Top

$E_{r}={\frac {M+2M^{n}}{2}}$, which simplifies to:

$E_{r}={\frac {M}{2}}+M^{n}$

Exhaust Velocity of Propellant

$V_{e}=I_{sp}\times g_{0}$, where

$V_e$ is the exhaust velocity of your propellant, and

$g_0$ is the standard gravity, which on Earth  is: 

$9.80665m/s^{2}$

Thrust to Weight

${\frac {F}{M_{g}}}\geq 1$, where

• $F$

is force of thrust in kilonewtons (kN) and

• $M$

is your rocket's mass in metric tons (t).

$g$ is the force of gravity. For example, on Earth , it is 9.80665 m / s ² .

If the TWR is greater than or equal to 1 your rocket will lift off the ground. 

Average ISP

${\bar {I}}_{sp}={\frac {1}{g_{0}}}\times {\frac {F_{total}}{{\dot {m}}_{total}}}$, where

• $F_{total}$ is the sum of all the forces, and

• $\dot{m}_{total}$is the fuel consumption rate in kg / s.

Average ISP can also be calculated from each engine:

${\bar {I}}_{sp}={\frac {F_{total}}{\sum _{n=1}^{E}{\frac {F_{n}}{I_{sp\,n}}}}}$, where

• $F_n$is the force in kilonewtons of the nth engine, including duplicates,

• $I_{sp\,n}$is the specific impulse of the nth engine, and
• $E$ is the number of engines.

Fuel Consumption Rate

The same fuel flow rate.

$rate=g_{0}\times {\frac {F}{I_{sp}}}$

Burn Time

${\frac {M_{f}}{\dot {m}}}$

Mass for that Time

$M_{f}-{\dot {m}}*time$

Suicide burn

$V_{f}=V_{i}+V_{e}\times \ln {\left({\frac {m_{i}}{m_{f}}}\right)}$

The height D at which you should start your burn from sea level is

$D=\left({\frac {V_{i}+{\sqrt {2\times g\times {D_{i}}}}}{2}}\right)\times T$, where T is the time it will take to burn, $T={\frac {\left(m_{i}-{\frac {m_{i}}{e^{\left({\frac {V_{i}+{\sqrt {2\times g\times D_{i}}}}{g_{0}\times I_{sp}}}\right)}}}\right)}{\left({\frac {F_{T}}{g_{0}\times I_{sp}}}\right)}}$

Where g is the surface gravity of the body upon which you're landing, D_i is initial altitude, V_i is initial vertical velocity, m_i is initial mass (with fuel) and F_T is thrust. See this Desmos calculator.

Orbital Velocity

$V={\sqrt {\frac {M}{r}}}$

Orbital Period

$T=2\times \pi \times {\sqrt {\frac {r^{3}}{M}}}$

Kepler's Third Law of Planetary Motion

$\left({\frac {Gmt^{2}}{4\pi ^{2}}}\right)^{3}$

SOI Radius

$a\left({\frac {m}{M}}\right)^{\frac {2}{5}}$

Max Payload 

$R=I_{sp}\times G\times \ln {\left({\frac {M_{f}+x}{M_{e}+x}}\right)}$

R = Required Δv for mission, x = Max Payload in tonnes

Eccentricity

$\xi ={\frac {r_{max}-r_{min}}{r_{max}+r_{min}}}$

$\xi$ = Eccentricity

r_max = Apoapsis

r_min = Periapsis

Please allow a margin of error of 10% as well as user errors.