Mathematical Question Calculating dV

[arobie]

Simply put, How do I calculate the dV of an orbital maneuver? For instance to start simply, hypothetically I have a rocket is in a 300 km X 300 km orbit. I want to increase the orbit to 300 km X 400 km. How do I calculate how much delta velocity I will need to achieve with my burn to get there?

 

[spacester]

I do this calc using orbital energy but other folks tend to use simpler formulas from textbooks, formulas that were not on the web when I did the self-teach thing on orbital mechanics in um 2000-2001. Assuming you have no rush for an answer, I'd just as soon defer and let someone else post those formulas and methods which more people are familiar with.<br /><br />Hint: 'Vis-Viva Equation' <div class="Discussion_UserSignature"> </div>

 

[strandedonearth]

Quick guess from someone who flunked out of the second year of engineering:<br /><br />Since your perigee remains the same, you should simply need to calculate your velocity at perigee (Vp) in the new orbit. Since your old orbit did not have a perigee, or was all perigee, the new Vp minus the old Vp should give your delta-v.<br /><br />But don't take my word for it, I'm sure there's a more expert way that will work for more orbital maneuvers, with differing perigees.

 

[search]

Have fun and do not burn to much...<br /><br />http://www.cdeagle.com/ommatlab/maneuvrs.pdf#search="calculate%20dV%20of%20an%20orbital%20maneuver"

 

[strandedonearth]

Well, it was just a quick guess by an amateur...

 

[spacester]

Wow. Is it that no one else knows how to do this, or that no one can be bothered? Or what?<br /><br />I won't be able to get to this until tonight, so I hold out hope that further chastisement will not be forthcoming. <img src="/images/icons/wink.gif" /> <div class="Discussion_UserSignature"> </div>

 

[propforce]

Hey Spacester,<br /><br />I thought you quit SDC? <img src="/images/icons/laugh.gif" /> <div class="Discussion_UserSignature"> </div>

 

[MeteorWayne]

He finds us irresistable.<br />Just had to chill out for a bit, I suspect.<br /> <div class="Discussion_UserSignature"> <p><font color="#000080"><em><font color="#000000">But the Krell forgot one thing John. Monsters. Monsters from the Id.</font></em> </font></p><p><font color="#000080">I really, really, really, really miss the "first unread post" function</font><font color="#000080"> </font></p> </div>

 

[spacester]

Yep, I had to chill out. I was a bad spacester.<br /><br />But mostly I'm addicted, simple as that. Like just now, instead of checking on this thread I'm supposed to be doing something else.<br /><br />Plus, someone has to do the math <img src="/images/icons/wink.gif" /><br /> <div class="Discussion_UserSignature"> </div>

 

[propforce]

I know what you mean... a mind is a terrible thing to waste.<br /><br />Plus, you have to do that math !! <img src="/images/icons/laugh.gif" /> <div class="Discussion_UserSignature"> </div>

 

[henryhallam]

Conservation of angular momentum: VpRp=VaRa where Vp=velocity at perigee, Rp=radius of perigee etc<br /><br />so Vp=VaRa/Rp<br /><br />also conservation of energy, kinetic+gravitational potential = constant. i.e. same at apogee and perigee.<br /><br />so 0.5Vp^2-GM/Rp=0.5Va^2-GM/Ra<br />where GM is the product of the universal gravitational constant and the planet's mass. For Earth it is about 3.99E+14 m^3/s^2.<br /><br />Solve the equations simultaneously to find Vp in terms of Ra and Rp. It's been a long day and I'm a little tired so I may have made a mistake but I get<br /><br />Vp=sqrt(2GM(1/Rp-1/Ra)/(1-Rp^2/Ra^2))<br /><br />Remember Rp and Ra are the radius of the perigee and apogee <i>from the centre of the Earth</i> not altitudes.<br /><br />Then simply calculate the perigee velocity for the final orbit and subtract the velocity for the starting orbit at the point of transfer (which could be perigee or apogee, in a circular orbit such as your example both are the same). The difference gives you the instant-acceleration delta-V. In reality the burn does not take place instantly so there will be gravity losses but these should be only a relatively small percentage of the total for most orbital transfers between nearly-circular orbits if the burn time is short.<br /><br />For your example of 300km circular to 300x400km I get inital velocity 7729.7m/s and final velocity 7758.4m/s for a dV of 28.7. This sounds vaguely plausible to me so the formula might be correct <img src="/images/icons/wink.gif" /><br /><br />To transfer between two circular orbits in the same plane the most efficient trajectory is a Hohman transfer, e.g. to go from 300x300 to 600x600 you should make two burns:<br />300x300 - /> 300x600<br />then wait until apogee of the elliptical transfer orbit<br />then 300x600 - /> 600x600.<br />So you would have to use that formula twice to work out the total dV. <br /><br />Spacester: you're not allowed to cop out of doing any maths so derive us the formula/e for a plane ch

 

[spacester]

Well done Henry! That's a heck of a challenge IIRC. Can I give you a little personal background first? The following reveals more personal information than the sum total of such since my start here, IOW it's a totally new thing for me, but it seems like it is time to do so. My apologies for what may seem like an indulgence but all things considered it seems like the right thing to do at this point in time. Yes, this is an evasion of sorts, so allow me to explain myself.<br />**<br />I am a Mechanical Engineer, and while I got mostly 'A's in calculus, diff. equations and applied linear algebra (matrix theory), baby that was my limit. That last one is the 'A' I am most proud of, but I had to really bear down to get it and I came away from that convinced that I did not have the capability to go higher, even if I had been required to or was interested in doing so. But yet, I'd always liked math, so my interest in space flight gave me an obvious direction. (BTW, I started College in 1976 and finished in 1995, with 13 years of making a living in between.)<br /><br />Back in 2001 I began an experiment with myself as the sole subject. I wanted to find out how much 'space math skill' a person (myself) could acquire by using <b>only the web.</b> No books were allowed, the idea was to find out what was on the web, identify what was missing and maybe down the road I could possibly fill in a gap or two myself.<br /><br />There were two types of questions that came up here regularly: rocket equation stuff and orbital delta V stuff. If there had been anybody else here at uplink willing and able to answer these questions, it is likely that I would have taken a different approach. As the resident optimist, the nature of the partial answers that were posted bothered me: they always seemed to say one or both of the following. A: Leave this difficult stuff to the experts, you silly person, or B: It cannot be done, space flight is HARD, you silly person, don't you know that? These were totall <div class="Discussion_UserSignature"> </div>

 

[spacester]

BTW, I found the 'textbook' formulae for transfer between eccentric orbits:<br /><br />Vp = sqrt [((2/Rp) - 1/a) * GM]<br />Va = sqrt [((2/Ra) - 1/a) * GM]<br />Where<br />Vp = Perigee Velocity<br />Va = Apogee Velocity<br />Rp = Orbital radius at Perigee = Re + Perigee Altitude<br />Ra = Orbital radius at Apogee = Re + Apogee Altitude<br />Re = Earth's radius = 6378.1 km (Earth at equator)<br />a = semi-major axis = (Ra + Rp) / 2<br />GM = 398605 km^3 / sec^2 (Earth)<br /><br />The Hohmann transfer orbit between eccentric orbit 1 and a higher eccentric orbit 2 has:<br />a,t = (Rp,1 + Ra,2) / 2<br />Vp,t = Perigee Velocity, transfer orbit<br />Va,t = Apogee Velocity, transfer orbit<br />Vp,t = sqrt [GM * ((2/Rp,1)-1/a,t)]<br />Va,t = sqrt [GM * ((2/Ra,2)-1/a,t)]<br /><br />dV1 = Vp,t - Vp,1<br />dV2 = Va,2 - Va,t<br /> <div class="Discussion_UserSignature"> </div>

 

[arobie]

Sorry, I've been busy, busy, busy lately, 15 hour days, but I wanted y'all to know that I have been checking in here for a few free moments and appreciate the help! Thanks SEARCH for that link, henryhallen for the math, and spacester for the math as well. When I get a chance, I'll absorb it into my knowledge, maybe in a day or two when I have time, and (!) I will have time because tomorrow morning I'm having my wisdom teeth removed, all four. I'll be out of school and work until Monday.<br /><br />So let's see the plane change maneuver math! Judging from SEARCH's link, it looks complicated, but so does everything else at first.<br /><br />Alright, gotta go to school.

 

[barrykirk]

I'm going to need to get back to this thread and examine the math in detail when... when I have time.<br /><br />I won't be happy until I fully grok it.

 

[arobie]

I've got another moment before I go to work, reading through this thread again, the math parts and spacester's long post especially. I don't have time but I did want to make a comment.<br /><br />spacester, nice post. That is exactly who you are from what I've gathered. Your explanations fit optimistic reasoning and your actions.<br /><br />Also, Oops, this morning, when I read your post at the end when you posted your math for plane change, I saw it, thought, "Awesome the math for plane changes!", but when I typed my post I completely forgot about seeing it. Oops! Thanks for posting your derivation of it.