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So the good ship Polaris has to have engine(s) and enough propellant to manage a total deltaV of 39,528 m/s.
Rick Robinson's Rocketpunk Manifesto blog has some important points to make. The performance of available rocket engines will affect the rate of exploration, what destinations can be reached, and the travel time.
Eric Rozier has an on-line calculator that will assist with some of these equations.
In order to calculate the spacecraft's total deltaV capacity, you need to know two things: the spacecraft's Mass Ratio, and the exhaust velocity of the engine. Surprisingly, you don't need to know anything else, not even the ship's mass.
Mass Ratio tells the percentage of the spacecraft's mass that is propellant. You generally try different values for the mass ratio until you get a deltaV that is sufficient. You want a mass ratio that is low, but you'll probably be forced to settle for a high one. As a rule of thumb, a mass ratio greater than 4 is not economical for a merchant cargo spacecraft, mass ratio 15 is at the limits of the possible, and a mass ratio greater than 20 is probably impossible (At least without staging. But we won't go into that because no self-respecting Space Cadet wants to go into space atop a disintegrating totem pole. For purposes of illustration, the Apollo Saturn V uses staging, and had a monstrous mass ratio of 22).
As a side note, propellant is also called "reaction mass" or "remass". Please note, there is a difference between propellant and fuel. Fuel is the material used by the propulsion system to generate energy. Propellant is "reaction mass", i.e., what comes shooting out the exhaust nozzle to work the magic of Newton's law of action and reaction. Only in rare cases (like chemical propulsion) are propellant and fuel the same thing. For most of these propulsion systems the fuel is uranium or plutonium and the propellant is hydrogen.
You probably won't use this equation, but the definition of mass ratio is:
R = M / Me
R = (Mpt / Me) + 1
- R = mass ratio (dimensionless number)
- M = mass of rocket with full propellant tanks (kg)
- Mpt = mass of propellant (kg)
- Me = mass of rocket with empty propellant tanks (kg)
If the Star Spear carries 70 metric tons of propellant, and the rocket masses 40 metric tons with dry tanks, its mass ratio is (70 / 40) + 1 = 2.75. This means that for every ton of rocket and payload there is 2.75 tons of propellant. Alternatively, if the Star Spear masses 110 metric tons full of propellant and 40 metric tons empty, the mass ratio is still 110 / 40 = 2.75. Note that mass ratios are generally always much higher than 1.0.
The equation you will actually use (later) is:
Pf = 1 - (1/R)
- Pf = propellant fraction, that is, percent of rocket mass that is propellant: 1.0 = 100% , 0.25 = 25%, etc.
The Star Spear's propellant fraction is 1 - (1 / 2.75) = 0.63 or 63%
The engine and its type determine Exhaust velocity. Often instead of exhaust velocity your source will give you an engine's "specific impulse". This can be converted into exhaust velocity by
Ve = Isp * 9.81
- Isp = specific impulse (seconds)
- Ve = exhaust velocity (m/s)
- 9.81 = acceleration due to gravity (m/s2)
Generally you will find the exhaust velocity (or specific impulse) of a given propulsion system listed in some reference work. I have a table of them here. It is possible to calculate the theoretical maximum of a given propulsion system, but it is a bit involved. I have a few notes for those who are interested, those who are not can skip to the next section. I'm only going to mention thermal type propulsion systems, non-thermal types like ion drives are even more involved.
For thermal type rockets:
Ve = sqrt( (2 * E) / m )
- Ve = exhaust velocity (m/s)
- E = energy (j)
- m = mass of fuel (kg)
Remember Einstein's famous e = mc2? For our thermal calculations, we will use the percentage of the fuel mass that is transformed into energy for E. This will make m into 1, and turn the equation into:
Ve = sqrt(2 * Ep)
- Ep = fraction of fuel that is transformed into energy
- Ve = exhaust velocity (percentage of the speed of light)
Multiply Ve 299,792,458 to convert it into meters per second.
|Particle||Mass (unified atomic mass units)|
Nuclear fission thermal rockets use uranium or other fissionables for fuel, and hydrogen for propellant. The fraction of the fissionables mass converted into energy is about 7 x 10-4. Ve = sqrt(2 * 7e-4) = 0.037 = 3.7% c. This is about 11,217,200 meters per second!
Remember, this is the theoretical maximum, in practice the exhaust velocity will be nowhere near this good. For one thing this simplistic analysis ignores a host of engineering problems, such as engine thermal limits. As an example, an actual design for a nuclear thermal gas-coaxial rocket has a calculated exhaust velocity of a mere 17,658 m/s, which is about 600 times as weak as the theoretical maximum.
Deuterium-Tritium Fusion rockets use the fusion reaction D + T ⇒ 4He + n. If you add up the mass of the particles you start with, and subtract the mass of the particles you end with, you can easily calculate the mass that was converted into energy. In this case, we start with one Deuteron with a mass of 2.013553 and one atom of Tritium with a mass of 3.015500, giving us a starting mass of 5.029053. We end with one atom of Helium-4 with a mass of 4.001506 and one neutron with a mass of 1.008665, giving us an ending mass of 5.010171. Subtracting the two, we discover that a mass of 0.018882 has been coverted into energy. We convert that into the fraction of fuel that is transformed into energy by dividing it by the starting mass: Ep = 0.018882 / 5.029053 = 0.00375.
Plugging that into our equation Ve = sqrt(2 * 0.00375) = 0.0866 = 8.7% c.
Deuterium-Helium3 Fusion rockets use the fusion reaction D + 3He ⇒ 4He + p. Start with one Deuteron with a mass of 2.013553 and one atom of Helium 3 with a mass of 3.014932, giving us a starting mass of 5.028485. We end with one atom of Helium-4 with a mass of 4.001506 and one proton with a mass of 1.007276, giving us an ending mass of 5.008782. Subtracting the two, we discover that a mass of 0.019703 has been coverted into energy. Ep = 0.019703 / 5.028485 = 0.00392.
Plugging that into our equation Ve = sqrt(2 * 0.00392) = 0.0885 = 8.9% c.
Hydrogen - Boron Thermonuclear Fission rockets use the reaction p + 11B ⇒ 3 x 4He. Start with one Proton with a mass of 1.007276 and one atom of Boron with a mass of 11.00931, giving us a starting mass of 12.016586. We end with three atoms of Helium-4, each with a mass of 4.001506, giving us an ending mass of 12.004518. Subtracting the two, we discover that a mass of 0.012068 has been coverted into energy. Ep = 0.012068 / 12.016586 = 0.001.
Plugging that into our equation Ve = sqrt(2 * 0.001) = 0.045 = 4.5% c.
Finally it is time to calculate the spacecraft's total DeltaV. For this, you can thank Konstantin Tsiolkovsky and the Tsiolkovsky rocket equation. This equation is the sine qua non of rocketry, without it this website would not exist. Sir Arthur C. Clarke called the most important equation in the whole of rocketry. If you are a serious rocket geek, you should have Tsiolkovsky's portrait hanging on your wall and the rocket equation on your T-shirt.
Anyway, the equation is:
Δv = Ve * ln[R]
- Δv = ship's total deltaV capability (m/s)
- Ve = exhaust velocity of propulsion system (m/s)
- R = ship's mass ratio
- ln[x] = natural logarithm of x, the "ln" key on your calculator
Suppose that the Polaris has a 1st generation Gaseous Core Fission drive. Exhaust velocity of 35,000 m/s (see table in engine list).
Let's try a mass ratio of 2 (50% propellant). 35,000 * ln = 24,260 m/s. Not good enough, we need 39,528 m/s.
Let's try a mass ratio of 3.1 (68% propellant). 35,000 * ln[3.1] = 39,600 m/s. That'll do.
There is a very important consequence that might not be obvious at first glance. What it boils down to is that if the delta V requirements for the mission is less than or about equal to the exhaust velocity, the mass ratio is modest and large payloads are possible. But if the delta V requirements are larger than the exhaust velocity, the mass ratio rapidly becomes ridiculously expensive and only tiny payloads are allowed. Most of the ship will be propellant tanks.
As a rule of thumb, the maximum economic mass ratio is about 4 (if the exhaust velocity of the engine cannot be changed, the optimum mass ratio is about 4.95). For such a mass ratio, the delta V requirement can be no larger than about 1.39 times the exhaust velocity of the propulsion system in question. Δv / Ve = ln = 1.39... (delta V no larger than 1.5 times the exhaust velocity if mass ratio is 4.95)
Turning it around, this means once you choose a propulsion system, you will know that it will not be able to do a mission with a delta V requirement over Ve * 1.39, not if you want to keep the mass ratio below 4 (or Ve * 1.5 if fixed Ve) .
Turning it around again, if you have chosen the mission, once you know the delta V you can calculate the optimal exhaust velocity. Ve = Δv * 0.72 (or Δv * 0.67 if fixed Ve)
Why is there an optimum value? If the exhaust velocity is too high, you are wasting energy in the form of high-velocity exhaust. If the exhaust velocity is too low, you are wasting energy by accelerating vast amounts of as-yet unused propellent. Dr. Geoffrey A. Landis says that this optimization is somewhat tedious to prove mathematically, you have to use calculus to maximize the value of kinetic energy of payload as a function of exhaust velocity.
In the real world, multi-stage rockets use a low exhaust velocity/high thrust engine for the lower stages and high exhaust velocity/low thrust engines in the upper stages.
Refer to the chart above to see how quickly the mass ratio can spiral out of control. Divide delta V by exhaust velocity and find the result on the bottom scale. Move up to the green line. Move to the left to see the required mass ratio. For instance, if the delta V requirement is 105,000 m/s, and you are using Gas Core rockets with an exhaust velocity of 35,000 m/s, the ratio is 3. Find 3 on the bottom scale, move up to the green line, then move to the left to discover that the required mass ratio is a whopping 20!
The inverse of the deltaV equation sometimes comes in handy.
R = e(Δv/Ve)
- e = 2.71828...
As a matter of interest, if the mass ratio R equals e (that is, 2.71828...) the ship's total deltaV is exactly equal to the exhaust velocity. Depressingly, increasing the deltaV makes the mass ratios go up exponentially. If the deltaV is twice the exhaust velocity, the mass ratio has to be 7.4 or e2. If the deltaV is three times the exhaust velocity, the mass ratio has to be 20 or e3.
These numbers are absolute, Mother Nature doesn't allow fudging. If your ship has a mass ratio of X and an exhaust velocity of Y, it will have a deltaV of Z. If the mass ratio is decreased due to the extra mass of, say, a stowaway, the deltaV goes down. If it goes down below what is needed for the mission, this signs the death warrant for everybody on board. Period. For details see the movie Destination Moon, or the short story "The Cold Equations" by Tom Godwin.
Now, remember that the percentage of the rocket mass that is taken up by propellant is:
Pf = 1 - (1/R)
This means that the percentage of the rocket mass that is not taken up by propellant is:
Pe = 1 / R
- Pe = percentage of rocket mass not take up by propellant
In other words, the rocket's dry mass expressed as a percentage of the rocket's wet mass. Substituting the equation for R we get:
Pe = 1 / e(Δv/Ve)
Pe is for the percentage of mass taken up by the propulsion system, the ship's structure, the payload, and anything else (like the crew). But hopefully most of Pe is payload, at least if this is a cargo ship. So given the ship's Δv capacity and the propulsion systems Ve, you can get a ballpark estimate of the ship's payload capacity.
This graph is the same as the previous one, only the vertical axis has be re-labeled to show how rapidly your payload shrinks (the other graph was labeled to show how rapidly the amount of propellant grows, which is more or less the same thing). See how steep the curve is? That is an example of what they call "rising exponentially", which is science-speak for "gets expensive real quick". The graph was drawn with the equation R = e(Δv/Ve). See how (Δv/Ve) is raised next to the e? That's what is called an exponent, its what makes the curve rise exponentially. This is why you want the delta-V to be as low as possible and the exhaust velocity to be as high as possible.
So what it is saying in English is that as the delta-V cost for the mission rises, the amount of allowed payload rapidly dwindles to zero. And using a rocket engine with a higher exhaust velocity will help. You lower delta-V by choosing more modest missions and/or using orbital propellant depots. You raise the exhaust velocity by using a more sophisticated engine.
To get some rough ballpark estimates, you can use my handy-dandy DeltaV nomogram (more about nomograms). Download it, print it out, and grab a ruler or straightedge. You can also purchase an 11" x 17" poster of this nomogram at . Standard disclaimer: I constructed this nomogram but I am not a rocket scientist. There may be errors. Use at your own risk.
Say we needed a deltaV of 36,584 m/s for the Polaris, that's in between the 30 km/s and the 40 km/s tick marks on the DeltaV scale, just a bit above the mark for 35 km/s. The 1st gen Gas Core drive has an exhaust velocity of 35,000 m/s, this is at the 35 km/s tick mark on the Exhaust Velocity scale (thoughtfully labeled "NTR-GAS-Open (H2)"). Now, lay the straightedge between the NTR-GAS-Open tick mark on the Exhaust Velocity scale and the "2" tick mark on the Mass Ratio scale. Note that it crosses the DeltaV scale at about 24 km/s, which is way below the target deltaV of 36,584 m/s.
But if you lay the straightedge between the NTR-GAS-Open tick mark and the "3" tick mark, you see it crosses the DeltaV scale above the target deltaV, so you know that a mass ratio of 3 will suffice.
The scale is a bit crude, so you cannot really read it with more accuracy than the closest 5 km/s. You'll have to do the math to get the exact figure. But the power of the nomogram is that it allows one to play with various parameters just by moving the straightedge. Once you find the parameters you like, then you actually do the math once. Without the nomogram you have to do the math every single time you make a guess.
As with all nomograms of this type, given any two known parameters, it will tell you the value of the unknown parameter (for example, if you had the mass ratio and the deltaV, it would tell you the required exhaust velocity).
Note that the Exhaust Velocity scale is ruled in meters per second on one side and in Specific Impulse on the other, because they are two ways of measuring the same thing. In the same way, the Mass Ratio scale is ruled in mass ratio on one side, and in "percentage of ship mass which is propellant" on the other.
Arthur Harrill has made a nifty Excel Spreadsheet that calculates the total deltaV and other parameters of your rocket.
For fun, you can spend $15 and get the RAND Rocket Performance Calculator, which is a circular slide rule for deltaV calculations. Its a pity it doesn't do metric, and the upper limit of Isp that it will handle is disappointing. But it does give one an intuitive feel for these calculations. (Alas, it appears that this is now out of print)
The General Electric Space Propulsion Calculator was manufactured by the GE Flight propulsion Laboratory. Front side calculates Thrust, Thrust Power, Propellant Mass Flow, Specific Impulse, and Exhaust Velocity. The flip side calculates Escape velocity, Orbital velocity, Period of revolution, and Gravitational pull for the major planets and moons of the solar system. Images are from the Slide Rule Museum. If anybody has any more information about this slide rule, please contact the webmaster.