Thrust-to-weight ratio

Thrust / Weight is the TWR of the rocket.

Thrust-to-weight ratio, abbreviated as TWR is a dimensionless ratio of thrust to weight of a rocket, jet or propeller engine or anything that provides work (thrust) and has a mass that is measurable.

Formula

The thrust-to-weight ratio (TWR) can be calculated by dividing the mass and thrust in any unit (e.g. newtons), then dividing the gravity to get the result (Earth's gravity is 9.81 m/s² (1 g)).

Rockets

For example: the mass of the rocket is 250 tons, and its thrust is 350 tons. It will lift off of Earth's gravity, so the gravity should be ignored. Divide the mass and thrust and you get this value:

$350t \div 250t = 1.4$

The TWR of this rocket is 1.4, which is a fairly good TWR. A good TWR is 1.3–1.5.

You should do this mathematics equation (thrust ÷ weight then TWR ÷ gravity of planet in g) to get the final TWR.

Now, another rocket has a mass of 150 tons and has a thrust of 400 tons. It will lift off of Earth's gravity as in the other example:

$400 \div 150 = 2.66666666667$

This rocket has a too high TWR, so it should decrease its thrust or increase its weight.

Another example: the rocket is on the moon. It has a mass of 27.5 tons and a thrust of 20 tons. If it lifts off of Earth's gravity, it will stay stuck. The moon's gravity is 1.62 m/s². So, the moon's gravity is 6 times weaker than Earth's so it has g = 0.1561.... Do the formula:

$20 \div 27.5 = 0.727 \div 0.1651 = 4.404...$

Now, we have a satellite in space that has a mass of 657 kilograms and a thrust of 1 ton (equivalent to roughly 1000 kilograms). If the rocket is in outer space, then we will calculate it using Earth's gravity. Calculate it and it has a TWR of 1.52, which can be seen in the formula:

$1000 \div 657 = 1.52$

Even if the TWR is less than 1, it will still accelerate.

A rocket is trying to escape Mars's gravity to get back to Mars orbit then get back to Earth. The rocket weighs 68.5 tons and has a thrust of 30 tons.

$30 \div 68.5 = 0.44 \div 0.3972 = 1.15$

It can lift off of Mars's gravity but it will slowly accelerate.

Now, try to have a rocket that has a mass of 320 tons and a thrust of 240 tons.

$240 \div 320 = 0.75$

This rocket has a TWR of less than 1. A TWR less than 1 means that the rocket cannot lift off of a planet. If it were moved to another planet with less gravity (e.g. Mars) it will lift off.

Now, try to calculate your rocket's TWR!

Engines

The TWR for engines can be calculated by the formula that can be seen above. Hawk engine for example:

$120t \div 3.5t = 34.29$

Apply it to other engines, such as the Frontier engine:

$100t \div 6t = 16.66666666667$

Titan engine:

$400t \div 12t = 33.33333333333$

Valiant engine:

$40t \div 2t = 20$

Kolibri engine:

$15 \div 0.5 = 30$

Try this in your own custom engines!

Average over operation

While thrust remains constant, payload mass decreases linearly with respect to time as fuel is burned. The average mass is thus $\frac{\int_{0}^{\frac{f_m}{c}} r_m+f_m-c*x dx} {\frac{f_m}{c}}=\frac{\frac{f_m (2r_m+f_m)}{2c}}{{\frac{f_m}{c}}}=r_m+\frac{f_m}{2}$.

Where rm is non-fuel rocket mass, fm is fuel mass, c is fuel consumption rate and x is time, so the thrust:(average weight over time) ratio is $\frac{t}{r_m+\frac{f_m}{2}}$.

where t is thrust. However, the average thrust-to-weight ratio over time is $\frac{\int_{0}^{\frac{f_m}{c}} \frac{t} {r_m+f_m-c*x} dx} {\frac{f_m}{c}}=\frac{\frac{t\ln{(\frac{r_m+f_m}{r_m})}}{c}}{{\frac{f_m}{c}}}=\frac{t\ln{(\frac{r_m+f_m}{r_m})}}{f_m}$.

Because $\Delta v=I_{sp}\ln{(\frac{r_m+f_m}{r_m} )}$ and $I_{sp}=\frac{t}{c}$, the average TWR is thus $\Delta v*\frac{c}{f_m}$, delta-v divided by operation duration.