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In my somewhat unrealistic school project regarding space tourism to Mars we have to accelerate a 500 ton module (mass of propellant and engines not included) from LEO to an orbit around the sun at Mars' distance. We have to transport all the fuel and propellants to the module ourselves.

When choosing engines, I'm not sure whether or not i should prioritize specific impulse or thrust to weight ratio. I want to calculate the propellant mass (and the amount of rockets) needed to achieve the desired delta v with the following equations.

Total impulse ($J_{tot}$):

$$J_{tot} = F_{avg} \cdot t$$

Engine burn time ($t$):

$$t = \frac{m_{propellant}}{\dot{m}}$$

Propellant flow rate ($\dot{m}$):

$$\dot{m} = \frac{F_{avg}}{I_{sp}}$$

Then:

$$\Delta p = m \cdot \Delta v$$

$$\Delta p = F \cdot \Delta t$$

My two questions are:

1: Regarding engine specifications, what's the difference between using their thrust to weight ratio or specific impulse as measurement of how effectively the engine achieve change in speed relative to their weight? I understand that the "weight" in thrust to weight ratio sometimes refers to the weight of the entire rocket so my spontaneous feeling is that in the specifications of engines, weight refers to the dry weight of the engine while specific impulse expresses the efficiency relative to propellant mass but I don't know. Should thrust to weight be prioritized (like Merlin 1D vacuum) or something with higher specific impulse?

2: How long can an engine burn fuel? Burn time is sometimes specified for engines but is it possible to prolong this? Are the engines' capabilities the limiting factor or is it the mass of propellant available? Or a little bit of both?

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I understand that the "weight" in thrust to weight ratio sometimes refers to the weight of the entire rocket so my spontaneous feeling is that in the specifications of engines, weight refers to the dry weight of the engine while specific impulse expresses the effectivity relative to propellant mass but I don't know.

When discussing an engine, the thrust-to-weight figure is always relative to the engine's weight, not the entire rocket's, and will be generally greater than 10, well over 100 in the case of some high-performance modern engines like the Merlin. This figure is relatively unimportant to overall performance, because the weight of propellant is much greater than the dry mass of a stage.

A rocket's overall thrust-to-weight ratio is much less often quoted. It will generally be less than 10 (except in a few very unusual cases). It usually starts at somewhere between 1.2 and 1.5 at liftoff, increasing over the course of a stage's burn, as propellant is consumed; it varies with payload and mission.

Should thrust to weight be prioritized (like Merlin 1D vacuum) or something with higher specific impulse?

In general, specific impulse is the more important figure, though high engine TWR also helps, all other things being equal.

How long can an engine burn fuel? Burn time is sometimes specified for engines but is it possible to prolong this? Are the engines' capabilities the limiting factor or is it the mass of propellant available? Or a little bit of both?

The burn times you see in specifications for a launcher are usually propellant-limited; they're the actual time the stage fires for on that particular launcher. Typically, engines and stages are co-developed, so the engine gets designed to run for as long as it needs to in that application. Some engines have ablative cooling (like the RS-68) so they're effectively destroying themselves at a controlled rate during use, and those can't generally be flown longer than their design times; some other engines are designed conservatively and can probably run a lot longer than rated. You can't tell just from a spec sheet; you need to research an engine's test history to find out if it's been run longer. In general, though, because of the diminishing-returns nature of the rocket equation, there's little point in extending burn times very much.

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Welcome to Space Stack Exchange! Great question.
ISP and Thrust to Weight ratio, in space, really aren't that important. Delta-Vee is important.
The questions that you are asking are answered by Delta-V. It is the amount of change in velocity (momentum) you can impart to the vehicle.
The total fuel capacity is, ultimately measured in Delta-V.
Fuel consumption is a little different in space than on the ground. In other words, I can burn half the fuel and take twice as long to get there.

Since you are already in LEO, you need to worry about the D-V to leave Earth Orbit, Enter the transfer orbit, enter Mars orbit, and possible, descend to Mars.
You will also need some Delta-V to maintain/'adjust your orbit.

That's the high level methodology to answer your question. (It's homework, not gonna give you all the answers).

To Summarize Compute your Detla-Vee's, then determine the engine that can provide that in a timely manner.

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    $\begingroup$ FYI: Lot's of folks try to focus on ISP. Really, that is nothing more than "Gas Mileage". How far can I go an a gallon of gas. Consider this question: "I am designing a semi that can carry a load across country. Should I go with the Engine that has the biggest horsepower, or go with the engine that has the best gas mileage?" The answer is: neither. $\endgroup$
    – Scottie H
    Oct 7 '20 at 18:17
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Partial answer to 2)

How long can an engine burn fuel? Burn time is sometimes specified for engines but is it possible to prolong this?

Chemical engines usually don't burn very long. There's an informative question here that gives some real world examples of "long" chemical rocket burns:

Longest continuous burning chemical rocket engine?

So, the vast majority are less than an hour. One burn is mentioned in comments that was more than two hours. I'd consider it an outlier.

Ion engines, however, can burn for very, very long times. Here's a ground test that ran for five and a half years. Real spacecraft like Dawn have very long burn times, this page says "weeks at a time".

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