Trip to Mars, ISP, mass fraction

Hi all

Lets say you are going to mars…there is some basic mass of junk you are going to have to take to breath, eat, do science, get into orbit, land, and take off to get back into orbit.

Now you vary that mass some depending on how fancy, redudant, or balls to walls risky you are…

Lets assume the minimum energy transfer to go to mars…the many month stay there…and the same slow minimum energy return…

The question I have is the equation for this:

What is the ratio of mass of payload of “junk” to mass of propellant used to get there as a function of ISP?

Thanks!

This is my area (more or less; I mostly work in structures but know a bit and some about propulsion) and I have to say that your question has few too many variables for a simple, explicit answer.

Backing up a bit, I[sub]sp[/sub] is known as specific impulse, or the ratio between the total impulse provided by the rocket to the mass of propellant burned. (For historical reasons and simplicity most industry planetary ballistics uses the total impulse divided by the surface weight of consumed propellant, giving a result in units of seconds, but the other definition is actually more correct.) Because the performance of a rocket will vary between full atmospheric pressure and vacuum (along with a few other parameters, particularly with solid fuel engines), the I[sub]sp[/sub] value stated in handbooks and promotional literature for orbital boosters is actually something of an average; propulsion gurus actually use thrust-time or impulse-time curves to predict payload capacity to a given orbit. Although specific impulse is widely used as a specific metric against which to compare different propellants in accomplishing the same goal (or as a general requirement for a mission of class such-and-such), for the question you are asking looking directly at the exhaust velocity, c is somewhat clearer.

For instance, assuming that we’re starting in vacuum from an Earth escape orbit, my back-of-envelope calculations say that you need any additional 2.9 km/s of velocity change to achieve a Hohmann (minimum energy) transfer. (There are other minimum energy transfers, but I don’t think that they require significantly less velocity change and their interval will be substantially longer). Once you get to Mars, you’ll need roughly 2.5 km/s of deceleration to circularize the orbit, giving a total delta velocity of about [symbol]D[/symbol]v=5.4 km/s. That delta velocity gives: [symbol]D[/symbol]v = c*[symbol]S[/symbol]ln(m[sub]i[/sub]/m[sub]f[/sub]) (the rocket equation)
Rearranging gives:c = [symbol]D[/symbol]v/ln(m[sub]i[/sub]/m[sub]f[/sub])
This gives you the exhaust velocity in terms of the overall mass fraction. (Of course, you are going to have to figure the return trajectory as well, but since we can assume that you’re going to leave a significant amount of your initial payload on the planet in terms of disposal lander components and fuel, equipment, et cetera, and only a modest amount of samples on the return journey, you’ll need to use a different payload mass for that leg.)

Now, if you want to convert that to I[sub]sp[/sub], then you’ll use the specific impulse equation, I[sub]sp[/sub]=c/gor alternatively, I[sub]sp[/sub]=c/(m[sub]i[/sub]-m[sub]f[/sub]) (which is, as alluded to above, more precise but less common in industry usage.)This gives an overall specific impulse requirement per mass difference, but doesn’t speak to total mass required. You can divide through by m[sub]f[/sub] to put the equation in terms of specific impulse per final mass in terms of exhaust velocity and mass fraction, but I find this a clunky and inelegant approach, especially since your mass fraction may change significantly with configuration.

If you are really interested in this topic, I recommend Prussing and Conway’s Orbital Mechanics (Chapter 5, pgs 85-87 has the relevant bits) as the clearest development of celestial mechanics I’ve found. Here is a decent page, too, albeit it throws it all at you at once.

Stranger

Thanks Stranger…just getting ready to go to bed…so not enough time to digest your post…
But lets make some simplifying assumptions…

operate only in vacuum…

going to mars with minimum energy transfer…the rocket used to put stuff into earth orbit first is something different that we are not interested in…and we arent interested in what takes you to the surface of mars or back up again either…

that the ISP is the ONLY thing that changes…or in otherwords no consideration of the rocket engine mass or the propellant tank mass even though if you had drastically different ISPs these things would probably be very different for such very different systems…

thanks again!

blll

I think that no matter which ISP you use, you will not get a very good broadband connection on Mars.

Ach! I missed a step; from the above post:[symbol]D[/symbol]m/m[sub]i[/sub] = 1 - m[sub]f[/sub]/m[sub]i[/sub] = 1 - e[sup]-[symbol]D[/symbol]v/c[/sup]giving you the propellant mass fraction in terms of the delta velocity requirement and exhaust velocity.

These are the essential assumptions I made with the above calculations. You can solve c from the previous post in terms of I[sub]sp[/sub] and initial and final vehicle mass, but the resulting equation is not linear with respect to one parameter now, so it becomes unpleasing. When comparing the efficacy of rocket propulsion systems I[sub]sp[/sub] is conventionally used as a means of relative comparison, but for figuring out mission requirements I would work from exhaust velocity and mass flow rate (not shown here because we’re idealizing thrust as a point-event, but relevant when you talk about actual thrust intervals), which will be readily scalable with mass fraction.

Stranger

Hi Stranger…
Thanks for taking the time to explain and type all that out! Unfortunately, without suitable reference material to put my grubby hands on, I am afraid that stuff is a bit beyond the mental effort I can/should devote to it at the time being…

But again, thanks!

Since you are in the “biz”, sorta, let me ask you a simpler yet related question…

How much delta V does it take to visit one of the near earth asteroids versus Mars?

And again thanks for the effort on your post here and the countless other informative post you have placed here over the years…

take care

Blll

The answer to that is “It depends.” (Aren’t you glad you asked these questions?) Although the Near Earth Objects are nearer than Mars; generally speaking, NEOs are objects that cross Earth’s orbit on a short period (~ 1 year) basis, roughly between 0.9AU and 1.3AU in distance. Some may come very close but are moving so quickly when they pass that it is prohibitive to try to catch them; others may meander around and require only a modest amount of delta velocity beyond that to escape a transfer orbit. For instance, at certain points in the orbit, the relatively velocity of 2002 AA[sub]29[/sub] and 3753 Cruithne is significantly less less than the ~4.0 km/s it takes to go from a geosynchronous transfer orbit to Lunar landing orbit. Of course, once you’ve got up to a transfer orbit, the amount of delta-v to go anywhere is pretty modest compared to what it takes to get there (~12-13 km/s), so if you have the energy and provisions to make a long journey it doesn’t cost you that much more than going a short way (as long as you don’t mind making elliptical coasting paths), which seems paradoxical to terrestrial travelers.

Looking back over the previous posts, I think I may have not made my point regarding specific impulse versus exhaust velocity entirely clear. Specific impulse is a property of the entire vehicle insofar as it includes the overall vehicle mass, so if you are developing a new vehicle it is not an independent parameter. Exhaust velocity, however, is (mostly) dependent only on the propellant used and energy throughput supplied. Given an exhaust velocity and the assumption that an engine can be scaled up or down linearly in size and power (not true, but close enough for a first order approximation) you can then use that is a fixed parameter to scale the rest of the system, i.e. payload, structure, fuel mass, et cetera. Trying to use I[sub]sp[/sub] for this would be like trying to figure out the resulting acceleration properties of a car knowing only the maximum horsepower of the engine; the fact is that there are a number of factors about the engine torque devleopment, transmission, vehicle mass, et cetera that will play into the ultimate acceleration capability of the car.

Stranger