Equations

{\displaystyle }

This is the master list for equations!

# v

I

s
p

×

g

0

×
ln
⁡

(

M

t

M

d

)

{\displaystyle \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.

### Top to Bottom

E

r

=
M
2

M

n

{\displaystyle 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

=

M
2

M

n

2

{\displaystyle E\_{r}={\frac {M+2M^{n}}{2}}}

, which simplifies to:

E

r

=

M
2

M

n

{\displaystyle E\_{r}={\frac {M}{2}}+M^{n}}

## Exhaust Velocity of Propellant

V

e

=

I

s
p

×

g

0

{\displaystyle V\_{e}=I\_{sp}\times g\_{0}}

, where

V
e

{\displaystyle V\_e}

is the exhaust velocity of your propellant, and

g
0

{\displaystyle g\_0}

is the standard gravity, which on Earth  is:

9.80665
m

/

s

2

{\displaystyle 9.80665m/s^{2}}

## Thrust to Weight

F

M

g

≥
1

{\displaystyle {\frac {F}{M\_{g}}}\geq 1}

, where

* F

{\displaystyle F}

is force of thrust in kilonewtons (kN) and

* M

{\displaystyle M}

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

g

{\displaystyle 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

I
¯

s
p

=

1

g

0

×

F

t
o
t
a
l

m
˙

t
o
t
a
l

{\displaystyle {\bar {I}}\_{sp}={\frac {1}{g\_{0}}}\times {\frac {F\_{total}}{{\dot {m}}\_{total}}}}

, where

* F

t
o
t
a
l

{\displaystyle F\_{total}}

 is the sum of all the forces, and

* m
˙

t
o
t
a
l

{\displaystyle \dot{m}\_{total}}

is the fuel consumption rate in kg / s.

Average ISP can also be calculated from each engine:

I
¯

s
p

=

F

t
o
t
a
l

∑

# n
1

E

F

n

I

s
p

n

{\displaystyle {\bar {I}}\_{sp}={\frac {F\_{total}}{\sum \_{n=1}^{E}{\frac {F\_{n}}{I\_{sp\,n}}}}}}

, where

* F
n

{\displaystyle F\_n}

is the force in kilonewtons of the *n*th engine, including duplicates,

* I

s
p

n

{\displaystyle I\_{sp\,n}}

is the specific impulse of the *n*th engine, and
* E

{\displaystyle E}

 is the number of engines.

## Fuel Consumption Rate

The same fuel flow rate.

r
a
t
# e

g

0

×

F

I

s
p

{\displaystyle rate=g\_{0}\times {\frac {F}{I\_{sp}}}}

## Burn Time

M

f

m
˙

{\displaystyle {\frac {M\_{f}}{\dot {m}}}}

### Mass for that Time

M

f

−

m
˙

∗
t
i
m
e

{\displaystyle M\_{f}-{\dot {m}}\*time}

## Suicide burn

V

f

=

V

i

V

e

×
ln
⁡

(

m

i

m

f

)

{\displaystyle 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

(

V

i

2
×
g
×

D

i

2

)

×
T

{\displaystyle 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

(

m

i

−

m

i

e

(

V

i

2
×
g
×

D

i

g

0

×

I

s
p

)

)

(

F

T

g

0

×

I

s
p

)

{\displaystyle 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, *Di* is initial altitude, *Vi* is initial vertical velocity, *mi* is initial mass (with fuel) and *FT* is thrust. See this Desmos calculator.

# V

M
r

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

# T
2
×
π
×

r

3

M

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

## Kepler's Third Law of Planetary Motion

(

G
m

t

2

4

π

2

)

3

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

## SOI Radius

a

(

m
M

)

2
5

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

# R

I

s
p

×
G
×
ln
⁡

(

M

f

x

M

e

x

)

{\displaystyle 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

# ξ

r

m
a
x

−

r

m
i
n

r

m
a
x

r

m
i
n

{\displaystyle \xi ={\frac {r\_{max}-r\_{min}}{r\_{max}+r\_{min}}}}

ξ

{\displaystyle \xi}

 = Eccentricity

rmax = Apoapsis

rmin = Periapsis

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