First a Mathematical Review

[arobie]

<b>First a Review</b><br /><br />It's been a while since I've done some of this stuff, and I feel like brushing up on it now, so a review for me. To anyone on these boards who knows this stuff, would you mind helping me out by double checking some of this for me?<br /><br /><i>The Rocket Equation</i><br /><br />Hmm let's see, I had all this written down in a folder, but I lost it when Katrina came through about a year ago. Time to relearn.<br /><br />So..<br /><br />mass fraction = e ^ dV / Ve<br />where<br />dV = Delta Velocity -- the change in velocity nessecary to reach one's target<br />Ve = Exhaust Velocity -- The velocity of the propellant shooting out of the rocket engine<br />Ve = g *Isp<br />g = 9.807 m / sec ^ 2<br />Isp = Specific impulse<br /><br />So that's how to find mass fraction, but what exactly is mass fraction?<br /><br />mass fraction is ratio of the original mass (mo) of the spacecraft to the final mass (mf) of the spacecraft.<br /><br />mass fraction = mo / mf<br /><br />The final mass is the mass of the spacecraft after the rocket burn. The original mass is the final mass plus the propellant mass.<br /><br />mo = mf + propellant mass<br />mass fraction = (mf + propellant mass) / mf<br />mass fraction = 1 + propellant mass / mf<br />mf * (mass fraction - 1) = propellant mass<br />propellant mass = mf * (mass fraction - 1)<br /><br />Ok, so those are my equations. I'm going to run through a scenario and toss some numbers in there. I have a ship in LEO destined for lunar orbit. No plane change is needed. It's transporting 30 tonnes of supplies. The dry mass (without propellant and payload) is 10 tonnes. The dV required to get to lunar orbit from LEO is ~3.9 km/s. The specific impulse of my rocket is 421 seconds.<br /><br />mass fraction = e ^ dV / Ve<br />dV = 3.9 km/s<br />Ve = 9.807 * 421 = 4128.747 m/s = 4.128747 km/s<br />mass fraction = 2.571775479<br /><br />propellant mass = mf * (mass fraction - 1)<br />mf = 40 tonnes<br />propellant mass = 40 * (2.571775479 - 1)<br />propellant

 

[arobie]

I was originally taught this math by spacester in his Hohmann Transfer to Mars Reference Thread a long while ago. So thanks to spacester for all that!<br /><br />Now I've got a question if anyone could help me with it.<br /><br />Towards the end of that thread, he asked me a question, "Why is the number [(1 / pf) - 1] important?" I was stumped at the time. I didn't truly understand the concepts although I was able to use the equations to solve for propellant. Now, I have a grasp of the concepts, but I'm still stumped by the question.<br /><br />I know that pf = Wp / GLOW<br /><br />Wp = mass of propellant<br />GLOW = Gross Life-Off Weight, the mass of the entire ships, payload, propellant, and all.<br /><br />(1 / pf) is just the inverse.<br />(1 / pf) = GLOW / Wp<br /><br />But I still don't know what (1 / pf) - 1 is. <br /><br />Can anyone offer me any help or advice here?

 

[spacester]

Hi Arobie! <img src="/images/icons/blush.gif" /><br /><br />Great to see you, nice math skills, thank you for the kind words and you give me too much credit.<br /><br />IIRC it was a stupid question on my part, if for no other reason than because I got the expression wrong. As I recall, I posted later with a correction. In any case, it was meant to lead you in the direction you took (congrats!) so it's not important now. I'm sorry if I've led you on a wild goose chase.<br /><br />To be honest with you, I'm not sure I'm even going to remember exactly what I was even getting at. Maybe it will come to me later.<br /> <div class="Discussion_UserSignature"> </div>

 

[arobie]

Ahh ok. Well thanks spacester.<br /><br />And no not too much credit. You taught the rocket equation to me when I had only completed algebra 1, and going by my first post on the thread when I didn't even sound like I commanded the intelligence to write and type properly. LoL. <br /><br />Now hmmm....I've got a question, but this one will need a new thread.