Delta V & transfer manoeuver : Doing the math

[Kraken1950]

Hello,

I have enjoy the game in Sandbox mode and now i'm starting a career of my own. If I am able to design rockets that can go to Mun & Minmus, land and come back safely, I am currently unable to produce rockets to go visit other planets without copying others' design. I'm fed up with this situation so I decided to do some math and get familiar with the Delta V and transfer manoeuver formulas. I hope I'll finally be able to design adequate spaceship without simply copying !

The problem is MechJeb is out with 1.05, therefore I can't check if my calculations are good. I'm posting this to ask for help and check if my method & results are good so I can do more math to travel on the edge of Kerbol system !

Here is a rocket I have designed, it can go to Mun or Minmus, orbit then come back home :

[IMG]http://image.noelshack.com/fichiers/2015/47/1448233138-orbit-2.jpg[/IMG]

Total mass is 43.2 t as shown on the picture. To calculate the Delta V I separated the rocket in 2 stages :

- 1st stage : 4xBACC + 1xFL-T800 + 1xLV-T30 + 4xTT-38K Radial Decoupler + 1x TR-18A Stack Decoupler for a total of 36.5 t ; fuel mass : 28.6 t ; dry mass : 7.6 t
- 2nd stage : 1xFL-T800 + 1xLV-909 + the rest for a total of 6.7 t

Given the example in the Wiki, I assumed that each stage must be calculate separatly i.e. first stage mass is 36.5 t and not the whole rocket for 43.2 t despite the first stage carry it all. Am I correct on this point ?

1. So 1st stage, using formula available in the wiki : g = 9.81 ; ISPaverage = 186.7012 (using the ATM value of engines 'cause atmosphere burn mainly) ; ln(36.5/7.6) my Delta V is 2873.985 m/s. If I consider that the total mass should prevail, I'm down at 1986.888 m/s of Delta V.

2. 2nd stage : Total mass = 6.7 t ; dry mass = 2.7 t ; ISP = 345 (vaccum value), my Delta V is 3075.977 m/s

TOTAL DELTA V FOR ORBIT 2 : 5949.962 m/s (or 5062.865 m/s if 1st stage is calculated with 43.2 t)

Am I correct ? I am using Excel sheet for calculation as I don't have a calculator.

For manoeuvre transfer from Kerbin to Mun, I'm using those values (again the formula come from the Wiki) :
r1 = 90 km
r2 = 120 000 km
R = 600 km
µ = 3530.461 km

I assume that I'm on a orbit around Kerbin at 90 km and I want to go to the Mun which is located at 120 000 km. Using the formula I obtain :

Dv1 = 0,852798 km/s ; I want m/s so I multiple by 1000 = 852,798 m/s which is consistent with the information available on the Wiki

If I want to know the Delta V for Duna, I just replace r2 by the Duna's altitude ==> 20 542 065 km.

[IMG]http://image.noelshack.com/fichiers/2015/48/1448235263-duna.jpg[/IMG]

Am I correct ?

Thank you in advance for your answers !

[sdrevik]

Google "Kerbin DeltaV map", and there's lots of maps showing transfers and insertions. They are optimistic / best case, so add 20% to 40%, depending on your skill.

Kerbin orbit is about 3500 to 4000 dV once you take into account atmosphere.

To calculate dV for craft, you may just want to install MechJeb just to get dV numbers. If you don't want to use MJ, just put the module on the ship in VAB and remove it before launch.

[juanml82]

Unless you love to do maths, install Kerbal Engineer Redux.

Roughly speaking, I'd say it's some 1,200 m/s dV for Duna or Eve (like a bit less, but just in case), some 1,000 m/s to capture if you can't aerocapture at the destination, some 1,500 m/sfor the return leg (it's likely less, but just in case) and some 1,000 m/s for capturing back at Kerbin, but you should look for the proper dV maps.

Going beyond Dres, you should calculate some 2,200 m/s for the ejection burn and capturing in moonless bodies without an atmosphere like Moho or Dres can take thousand of m/s

[Snark]

[quote name='Kraken1950']I am currently unable to produce rockets to go visit other planets without copying others' design. I'm fed up with this situation so I decided to do some math and get familiar with the Delta V and transfer manoeuver formulas. I hope I'll finally be able to design adequate spaceship without simply copying ! [/QUOTE]

Great, good for you! :)

Other posters are correct that you can save a lot of math by using a mod like KER. If you like doing math, you can figure it out yourself. (Full disclosure, I'm a physics and math nerd so prefer to do it myself and have never run KER.)

[quote name='Kraken1950']Total mass is 43.2 t as shown on the picture. To calculate the Delta V I separated the rocket in 2 stages :

- 1st stage : 4xBACC + 1xFL-T800 + 1xLV-T30 + 4xTT-38K Radial Decoupler + 1x TR-18A Stack Decoupler for a total of 36.5 t ; fuel mass : 28.6 t ; dry mass : 7.6 t
- 2nd stage : 1xFL-T800 + 1xLV-909 + the rest for a total of 6.7 t

Given the example in the Wiki, I assumed that each stage must be calculate separatly i.e. first stage mass is 36.5 t and not the whole rocket for 43.2 t despite the first stage carry it all. Am I correct on this point ?[/QUOTE]

No.

To do the math for a multi-stage rocket, it works like this:

For each stage,

1. Take the total (wet) mass of the rocket including the stage and payload
2. Divide by the "dry" mass which is the wet mass minus [I]only the fuel in that stage[/I]
3. Use that as your mass ratio to plug into the Tsiolkovsky rocket equation to get dV for that stage

Note that the starting wet mass of the next stage will be less than the ending dry mass of the previous stage, since you're dropping dead weight (which is the whole point of staging, after all).

You're making the math a bit complicated by having [I][U]one[/U][/I] stage that has mixed Isp (SRB very different Isp from Reliant), where the engines run out of fuel at different times. I could talk you through the process of figuring it out, but it would be somewhat ugly and complicated. Isn't there any way you can stage those SRBs separately? For example, even if you're still bumping up against a 30-part limit, you could put a stack decoupler under the Reliant, then have the SRBs mounted to the decoupler rather than the tank above. Thus your 1st stage would be just the SRBs, then the 2nd stage would be the Reliant.

[Streetwind]

Well, he has enough part allowance even then to add four radial decouplers, since he's only at 25. And his engineer's report says he has a 255 part allowance, so it's not an issue.


My question is more along the lines of: why [I]do[/I] you feel that you are unable to design ships for interplanetary travel? Perhaps you have not realized it yet, but the KSP solar system is tiny, the the dV costs are tiny as well. A transfer to Duna costs pretty much the same as a transfer to Minmus, maybe at most 100 dV more if you manage to avoid an expensive plane change for Minmus. The same is true for Eve. That means that a craft that is able to land on the Mun and return (which is more expensive than doing the same on Minmus) is perfectly capable of reaching Duna and returning, or reaching Eve and returning. Yes, the [I]exact same[/I] craft. Perhaps you might even be able to land on Gilly and still return to Kerbin.

Now, [I]landing[/I] on and returning from a planet like Duna is a bit more tricky (not to mention Eve, which is the hardest place to return from in the entire game). But if you're only concerned with getting there and back, then just take the vessel you already have and fly there. Get a feel for it. The process of doing so is its own topic, but since you're asking about shipbuilding only, I assume you have looked into that already.

Of course, your vessel can be improved. For starters, I'll echo the suggestions already made: install Kerbal Engineer Redux. Then, separate the solid boosters into their own stage, with radial decouplers. Next, I'll add my own suggestion: remove the LV-T30 and add a LV-T45; it's improved Isp will beat the lower weight of the former when you use it as a second stage. Also add a second FL-T800 for the LV-T45 stage - in fact, if its TWR is still well above 1 then, consider adding more fuel, as you have a large upper stage. Don't drop the TWR too low though, else you may have problems during the ascent. Finally, use KER's help to adjust the SRB throttles for a takeoff TWR of about 1.5 to 1.6, if they aren't already there.

[Yasmy]

I'll address your second question on how to calculate the cost of interplanetary travel.

First, there are some high quality delta-v maps around. Be careful to use an up to date map. While I've calculated everything myself from time to time, I still use delta-v maps when planning most of my missions.

Second: It appears you correctly calculated the first burn of a Hohmann transfer from low Kerbin orbit to the Mun, which is about 860 m/s.
Next you might want to calculate the second burn to circularize in Munar orbit, but the second Hohmann burn equation is not correct because it ignores the presence of the Mun.
The equations are [img]https://upload.wikimedia.org/math/1/b/5/1b54b322a2dd6656ea92162ac419bb28.png[/img] and [img]https://upload.wikimedia.org/math/d/7/f/d7f826b60051adf001ccd0f45159a618.png[/img] respectively.

Now you can use these equations for part of an interplanetary travel calculation, but it is a bit more complicated than a Hohmann transfer orbit between two orbits around a single body. First though, you need to use mu for Kerbol, not Kerbin, and r1 and r2 as the orbital radii of your starting and ending planets at time of departure and intercept.

So step 1 is calculating the Hohmann transfer burns. These are not the burns you will be performing. These are the burns you would make to go from one planet's orbit to another planet's orbit if the planets were not there. Next you have to account for the gravity of the planets. This is good, as the planets will actually help make the transfer cheaper.

delta-v1 is how fast you want to be going right before exiting Kerbin's SOI, so that your Kerbol-relative velocity will be Kerbin's orbital velocity plus delta-v1. Likewise delta-v2 is how fast you will be moving relative to your destination planet when you reach it.

Step 2 is to calculate how to exit Kerbin's SOI with velocity delta-v1, starting from some initial orbit around Kerbin. The easiest way is to use the [url=en.wikipedia.org/wiki/Vis-viva_equation][i]vis-viva[/i][/url] equation: [img]https://upload.wikimedia.org/math/4/0/6/4066c272245ac8455dde64dedf1d7bed.png[/img].

Now that we are back in Kerbin's SOI, use Kerbin's mu. While your velocity and radius change along an orbit, the semi-major axis is constant. We can use the vis-viva equation at two locations, exploiting this fact. At the SOI, the radius is the SOI radius, R, and the velocity is delta-v1. Now pick some initial low Kerbin orbital radius (call it r0) at which to perform your initial transfer burn.
1) (delta-v1)^2 = mu (2/R-1/a). Solve for the unknown semi-major axis: mu/a = 2mu/R - (delta-v1)^2.
2) Substitute the semi-major axis back into the equation at LKO: v1^2 = mu(2/r0 - 1/a) = mu(2/r0-2/R) + (delta-v1)^2
3) v1 is how fast you want to be going at LKO at radius r0, to ensure that you will be going delta-v1 at the SOI. Thus your first burn is delta-v0 = v1 - v0, where v0 is your initial LKO velocity. If your initial orbit is circular with radius r0, the initial orbital velocity is v0 = sqrt(mu/r0). (You can show this from the vis-viva equation.)

Now you know your transfer burn: delta-v0 = v1 - v0. Be careful to use the correct mu everywhere. Kerbol's in delta-v1 and delta-v2. Kerbin's in v1.

Step 3 is just like step 2. Given the Hohmann transfer arrival velocity delta-v2 at your destination planet's SOI, calculate the delta-v, delta-v3 needed to slow down to orbital velocity at some radius r3. The math is identical. Use the destination planet's mu in the vis-viva equation.

Step 4. The total transfer and injection delta-v is delta-v0 + delta-v3, though if your destination has an atmosphere, you can reduce delta-v3 by aerobraking.

(Aside: Some people burn so they just escape a planet, burn the first Hohmann transfer maneuver, (skip slowing down at the destination SOI) then burn to circularize in low orbit. For these maneuvers, the cost is (v_escape-v0) + delta-v1 + delta-v3, which is always higher than delta-v0 + delta-v3. You can prove this with the equations above. v_escape is the velocity needed at LKO to have zero velocity at the SOI, instead of delta-v1.)

P.S: Calculating ejection angles is a good bit trickier. It is just geometry, but it is a pain to get right. Writing that up would take me some time.
P.P.S: Calculating landing and liftoff delta-v is also way too much to cover in this single post.
P.P.P.S: LKO to the Mun: use delta-v1, the LKO to Munar orbit transfer burn plus delta-v3, the circularization inside a destination SOI burn to calculate the cost of transfer from LKO to the Mun. As always, watch your mus. Edited November 23, 2015 by Yasmy

[tomf]

It is from a whila ago but I went through an example of calculating delta-vs required for interplanetary transfer here and I think it is pretty well explained.
[url]http://forum.kerbalspaceprogram.com/threads/27171-Calculating-interplanetary-delta-v[/url]

Everything in that thread is still valid. However, now I have proved to myself that I can do it I almost always just use Alex Moons transfer planner.

[Kraken1950]

My primary goal is to understand what i'm doing and being able to anticipate my future space travel. When I’ll be familiar with the theory and calculation, I will sure save time & use a third party mod. But for now, it is math !!

Thanks to you, I see what I did wrong so I will recalculate the Delta V. Can you confirm I am correct for the 2nd stage ? This stage is used only in space.

[Quote] My question is more along the lines of: why do you feel that you are unable to design ships for interplanetary travel? Perhaps you have not realized it yet, but the KSP solar system is tiny, the the dV costs are tiny as well.[/Quote]

Indeed, I saw that for most planets or satellites you need between 6k to 8k Delta V but I feel I ‘m unable to design rockets because I don’t know what I’m doing : I don’t know how much I need and I don’t know how much delta V I’m building by putting all those fuel tanks & engines together. Of course, I can use mods and existing resources but where is the creativity and the wonderful process of learning ?

Thank you for your advice on the rocket design. However, when I check the values of the LV-T45, ISP is slighty inferior (270 compare to the 280 of the LV-T30) in atm but slightly better in space (320 > 300) and I get less thrust for a 0.25 t gain in mass (+ the mass of the extra tank fuel that you recommend). How it will help apart from burning longer ? I was using the LV-T30 ‘cause of alternator now that I have solar panel, I can reconsider it.

Yasmy

[Quote] I'll address your second question on how to calculate the cost of interplanetary travel.

First, there are some high quality delta-v maps around. Be careful to use an up to date map. While I've calculated everything myself from time to time, I still use delta-v maps when planning most of my missions.

Second: It appears you correctly calculated the first burn of a Hohmann transfer from low Kerbin orbit to the Mun, which is about 860 m/s.
Next you might want to calculate the second burn to circularize in Munar orbit, but the second Hohmann burn equation is not correct because it ignores the presence of the Mun[/Quote]

Thanks for your answer, lots of thing to process though ! What do you mean by “mu” Kerbol, what value is it ? Mass x Gravitational constant ?
Secondly, in the Hohmann transfer orbit formula why don’t we use R (radius of parent body) in this formula ? Once out os Kerbin SOI the R could be Kerbol ?
I will check on the vis-viva formula as do not understand the use right now.

Thank you all for your answers, it is a great help !

[Racescort666]

[quote name='juanml82']Unless you love to do maths, install Kerbal Engineer Redux.

*snip*[/QUOTE]

Has KER been updated for 1.0.5 yet? Last I checked it wasn't [totally] compatible. There is a Dev version out there if you want to go that route but it requires some more leg work to track down.

[juanml82]

[quote name='Racescort666']Has KER been updated for 1.0.5 yet? Last I checked it wasn't [totally] compatible. There is a Dev version out there if you want to go that route but it requires some more leg work to track down.[/QUOTE]

Most of it is compatible. I'm not sure if the new engines work, but everything else is ok

[Yasmy]

[quote name='Kraken1950']<snip> What do you mean by μ Kerbol, what value is it ? Mass x Gravitational constant ?
Secondly, in the Hohmann transfer orbit formula why don't we use R (radius of parent body) in this formula ? Once out os Kerbin SOI the R could be Kerbol ?
I will check on the vis-viva formula as do not understand the use right now.

Yes, by mu (the English spelling of the Greek letter μ), I mean the gravitational parameter, G*M. It's easier to type than looking up μ and cutting and pasting, if you don't know a keyboard shortcut. M should be the mass of the object you are currently orbiting.
In the Hohmann transfer orbit formulae, r1 and r2 are the distances from the center of the body you are orbiting at departure and arrival because these distances determine your gravitational potential energy. For interplanetary transfers, the central body is Kerbol, and r1 and r2 are approximately the orbital radii of the initial and final planets. Really they should be the spacecraft's Kerbol radii at the SOI boundaries, but the planetary orbital radii are a fabulous first approximation. The error is less than Kerbin's SOI radius divided by Kerbin's orbital radius: 8.42e7/13.6e12 = 6e-6. Completely ignorable. (For Jool the error can be on the order of 3%, since its SOI is so huge.)

The way I used the vis-viva equation is entirely equivalent to using conservation of energy:
E1 = 1/2 m v1^2 - GMm/r1
E2 = 1/2 m v2^2 - GMm/r2
E1 = E2, thus v2^2 = μ(2/r2-2/r1) + v1^2

Don't forget that in KSP, orbital altitudes are reported, which are distances from nominal planetary surface, rather than distances from the center of the planet, so you often have to add the planet's radius to its altitude to find r, the distance of the spacecraft from the center of a planet. So for transfer from LKO at 100 km altitude to geosynchronous orbit at 1581.76 km altitude, you would use r1 = 100 km + 600 km, and r2 = 1581.76 km + 600 km, since Kerbin's radius is 600 km. In contrast, the wiki lists planetary orbital information using semi-major axis, periapsis and apoapsis all measured relative to the center of Kerbol, not its surface.

Edited November 29, 2015 by Yasmy
forum migration craziness

[Streetwind]

[quote name='Kraken1950']Indeed, I saw that for most planets or satellites you need between 6k to 8k Delta V but I feel I ‘m unable to design rockets because I don’t know what I’m doing : I don’t know how much I need and I don’t know how much delta V I’m building by putting all those fuel tanks & engines together. Of course, I can use mods and existing resources but where is the creativity and the wonderful process of learning?[/QUOTE]

There are a few rocketry rules of the thumb you can use to guide yourself.

1.) Heavy bits at the front, draggy bits at the back. Like an arrow, where the arrowhead is metal and the feathers are all the way at the rear. This configuration makes rockets fly true, just like arrows. If you reverse this setup in your rocket build... then well, your rocket will seek to reverse itself in flight, and you will not go to space today.

2.) As you learned in this thread, for rocketry purposes, higher stages effectively become payload for lower stages. From the viewpoint fo your launch stage, the second stage and everything above it, including fuel, is all payload. All dry mass.

3.) Keeping this in mind, try building stages with similar fuel/mass fractions. If your highest stage up is 50% fuel by mass, then you should size your next stage down so that it, too, has just as much fuel by mass as its own dry mass [I]plus[/I] its payload, namely the entire fueled upper stage (as per rule 2). Note: I'm not implying that a 1:1 fuel/mass fraction is what you should build, I just used it because it makes the sentence easier to write and more clearly understandable to the reader :P What fuel/mass fraction you want/need is determined by how many stages you have, how much dV you need, and when precisely you need to stage certain things - when you decouple your lander is usually defined by being at the destination, for example, and not by trying to hit a fuel/mass fraction in the editor.

You can approximate rule 3 by trying to keep roughly the same dV in each stage, provided the Isp of your engines is roughly similar from stage to stage. Allow stages with higher Isp engines to err a bit towards higher dV.

You will encounter situations where your best course of action is to knowingly violate this rule. This is perfectly okay, so long as you put thought into what you are doing. Knowing the rule is the first step to learning how to break it :P

4.) As a corollary/variant to rule 3, it's a good idea to have the thrust of each next stage down be between 3 and 4 times that of the current stage. If you keep your TWR in comfortable territories, this will automatically result in fairly well-sized stages. For example, a Terrier in the upper stage is well partnered with a Swivel in the stage below it, due to 60 kN versus 200 kN fitting nicely in that "between 3 and 4 times" rule of the tumb. Not all engines of similar stack sizes in KSP work out that well, but it's only a rough guideline after all, not a hard law.

5.) You're allowed to stage in interplanetary space. ;) While a lot of people really enjoy building giant motherships that execute a whole interplanetary trip and return in a single stage, it doesn't mean you have to. If you'd rather still have four stages left while already in LKO, that works just as well, and makes it [I]a lot[/I] easier to produce large dV figures.

Hope that helps!



[quote name='Kraken1950']Thank you for your advice on the rocket design. However, when I check the values of the LV-T45, ISP is slighty inferior (270 compare to the 280 of the LV-T30) in atm but slightly better in space (320 > 300) and I get less thrust for a 0.25 t gain in mass (+ the mass of the extra tank fuel that you recommend). How it will help apart from burning longer ? I was using the LV-T30 ‘cause of alternator now that I have solar panel, I can reconsider it.[/QUOTE]

Atmosphere density falls exponentially with altitude. At 10 km up, you already only have 1/8th of sea level pressure to deal with. That means, the Isp of the LV-T45 'Swivel' will already be 7/8ths of the way towards 320... in other words, it will be past 313 and increasing, while the LV-T30 'Reliant' cannot even pass 300 in pure vacuum. And at an altitude of 10 km, like in this example, you're probably still riding your solid rocket motors and the liquid engine isn't even started yet. Carrying a Reliant to this altitude before staging it is a waste when you have the Swivel available, in addition to the Swivel's very helpful thrust vectoring. And both of those engines have generous alternators, so power won't be an issue.

Rockets start by going up vertically and leave the dense atmosphere very fast, making sea level Isp irrelevant after the first 20-30 seconds. Then rockets spend a lot of time accelerating sideways high up in the atmosphere, where performance is defined by vacuum Isp. By the time you pass 20 km, the Swivel will be pretty much at peak performance for all intents and purposes, and far outshines the Reliant. This counts double if the engine is pushing a lot of weight, since that will marginalize the mass difference between the engines. Very lightweight final stages are usually cases where going for the lighter engine gives better overall dV even with less Isp, but those are the exceptions to the rule.

[Kraken1950]

Thanks you all for your answers !