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Expanded dV map for planning lunar missions


sevenperforce
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I got tired of doing a complex set of calculations from scratch every time I wanted to find the dV of a given Hohmann transfer, so eventually I sat down and made an excel calculator to do it for me. Which led to this:

reference_table_earth.png

And this:

reference_table_moon.png

These are reference tables for periapse and apoapse velocities for Hohmann transfers between numerous orbits of interest around the earth and the moon. They should be pretty self-explanatory.

These won't give you dV directly; instead, you have to subtract your current velocity from your target velocity. So if you're at a low orbit and want to go up to a higher one, subtract your circular-orbit velocity (in purple) from the periapse velocity (in pink) matching your orbital altitude to the target altitude and execute that burn. Once you reach the apoapse of the Hohmann transfer at your target altitude, subtract your new velocity (in blue, matching the new altitude to your starting altitude) from your target orbit's circular velocity (in purple) and execute that burn to circularize. To drop to a lower orbit, do the same thing in reverse. 

The EML-1 and EML-2 points are reference for an orbit at that distance; the perigee burn is the same, but the apogee burn needs to match the lunar-circular velocity instead. Actually that's not perfectly correct (since it matches period not speed), but I'm using patched-conic anyway so it's close enough. I've made a correction to the original so that the EML-1 and EML-2 circular velocities are the period-matching velocities rather than the reference velocities for an orbit at that distance. On the lunar side those points are stationary so you don't have to match velocity at all, In the lunar reference table, the circular velocities for EML-1 and EML-2 are for orbits with that distance but at other points; if you are actually reaching one of these points, you just kill your elliptic-orbit apolune velocity. 

Then I decided to go ahead and create a complete dV map for all major cislunar transfers. I can't attach it here but I posted it at the following link:

http://forum.nasaspaceflight.com/index.php?topic=39942.0

Here's a reduced-size version of the dV map; if you want the full-size version, you'll have to go to the link above or click here.

Exhaustive_d_V_map_Earth_moon.png

Edited by sevenperforce
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20 minutes ago, Exoscientist said:

The colors you quote 'red' and 'blue' don't match the colors on my computer. What are the varying numbers supposed represent in a row?

 

  Bob Clark

On the top and far left is the distance in km from the point mass. The top and far left represent apses.

Let's say you want to know how fast you're ship would go at a certain apsis. The speed depends on the other apsis, and so you pick one, then the other, and read it.

For example, let's say my ship has one apsis at the edge of the hill sphere, and the other at Earth's surface. At the hill sphere edge it would have an orbital speed of 47 m/s. At the earth's surface it would have an orbital speed of 11156 m/s. 

That's the best I can figure.

OP, I think it might've been more prudent to use angular momentum in an excel sheet to just calculate the velocity at a certain distance. Once you have the orbital velocity of a certain apsis, of course.

Edited by Bill Phil
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9 hours ago, Exoscientist said:

The colors you quote 'red' and 'blue' don't match the colors on my computer. What are the varying numbers supposed represent in a row?

 

  Bob Clark

The colors are pink, purple, and pale blue; depending on your computer settings the pale blue may appear to have a greenish tinge. 

The italized axis labels represent varying apsides. When the apsis on the horizontal axis matches the apsis on the vertical axis, the orbit is circular and has only one speed; when the axes don't match, there is a high apsis (the apoapse/apogee) and a low apsis (the periapse/perigee). Because the axes have the same labels, each possible combination appears twice, once in the pink region and once in the blue region. But since this is now an elliptical orbit without constant speed, the minimum speed (at apoapse) is shown in blue while the maximum speed (at periapse) is shown in pink.

For most coplanar orbits, the most efficient way to transfer between circular orbits at different altitudes (for example, to go from a 100 km circular parking orbit to the 400 km circular orbit of the ISS) is to enter an elliptical orbit with a perigee at 100 km and an apogee at 400 km, ride it up to its highest point, then transfer into a circular 400 km orbit with a circularization burn. Virtually all orbital maneuvers involve some sort of elliptic transfer. This table allows you to figure out how much speed you need to enter or to leave any of the given orbits. 

9 hours ago, Bill Phil said:

OP, I think it might've been more prudent to use angular momentum in an excel sheet to just calculate the velocity at a certain distance. Once you have the orbital velocity of a certain apsis, of course.

That's the thing; you don't necessarily have the velocity at a given apsis. If I'm in a 185x185 km circular parking orbit and want to send a satellite to geostationary orbit, I won't know how large of a burn I need. But glancing at the table, I see that the perigee speed for a 185km x 36,786km elliptical orbit is 10,268 m/s. My parking orbit speed is 7,793 m/s (shown in purple). So I need 3,275 m/s of dV to enter the geostationary transfer orbit.

Once the satellite completes the transfer, I can look at the apogee speed for that same 185km x 36,786km orbit to figure out how much additional velocity the satellite will need in order to circularize to a 36,786x36,786 km geostationary orbit.

So it's not just about finding perigee speed for a given apogee (or vice versa); it's about calculating the energy requirements for every different transfer or maneuver. 

The dV map I linked to shows all this even more clearly; it was just too large an image to embed here. 

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15 hours ago, sevenperforce said:

I got tired of doing a complex set of calculations from scratch every time I wanted to find the dV of a given Hohmann transfer, so eventually I sat down and made an excel calculator to do it for me. Which led to this:

reference_table_earth.png

And this:

reference_table_moon.png

These are reference tables for periapse and apoapse velocities for Hohmann transfers between numerous orbits of interest around the earth and the moon. They should be pretty self-explanatory.

These won't give you dV directly; instead, you have to subtract your current velocity from your target velocity. So if you're at a low orbit and want to go up to a higher one, subtract your circular-orbit velocity (in purple) from the periapse velocity (in pink) matching your orbital altitude to the target altitude and execute that burn. Once you reach the apoapse of the Hohmann transfer at your target altitude, subtract your new velocity (in blue, matching the new altitude to your starting altitude) from your target orbit's circular velocity (in purple) and execute that burn to circularize. To drop to a lower orbit, do the same thing in reverse. 

The EML-1 and EML-2 points are reference for an orbit at that distance; the perigee burn is the same, but the apogee burn needs to match the lunar-circular velocity instead. Actually that's not perfectly correct (since it matches period not speed), but I'm using patched-conic anyway so it's close enough. I've made a correction to the original so that the EML-1 and EML-2 circular velocities are the period-matching velocities rather than the reference velocities for an orbit at that distance. On the lunar side those points are stationary so you don't have to match velocity at all, In the lunar reference table, the circular velocities for EML-1 and EML-2 are for orbits with that distance but at other points; if you are actually reaching one of these points, you just kill your elliptic-orbit apolune velocity. 

Then I decided to go ahead and create a complete dV map for all major cislunar transfers. I can't attach it here but I posted it at the following link:

http://forum.nasaspaceflight.com/index.php?topic=39942.0

Here's a reduced-size version of the dV map; if you want the full-size version, you'll have to go to the link above or click here.

Exhaustive_d_V_map_Earth_moon.png

I think I went blind reading this graph. Even on my 32 inch monitor I don't think I could read this.

I not sure if anyone has mentioned this by you have not give a dV from the moving earths surface to orbit. So for example if you launch from ESA site in S. AMerica, or canavaral, or russian lanuch site, etc. How much dV does it take to reach these various equitorial parking LEO orbits. Seems like that is the 800 lb elephant in the center of the room. Each site has a different amount of dV spent because of it latitude (slower surface velocity with increasing latitude and cosine loses for vector angles to equator and inclination correction losses.

 

 

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3 minutes ago, PB666 said:

I think I went blind reading this graph. Even on my 32 inch monitor I don't think I could read this.

Yeah, the graphic is very large. You can download the pdf version here.

5 minutes ago, PB666 said:

I not sure if anyone has mentioned this by you have not give a dV from the moving earths surface to orbit. So for example if you launch from ESA site in S. AMerica, or canavaral, or russian lanuch site, etc. How much dV does it take to reach these various equitorial parking LEO orbits. Seems like that is the 800 lb elephant in the center of the room. Each site has a different amount of dV spent because of it latitude (slower surface velocity with increasing latitude and cosine loses for vector angles to equator and inclination correction losses.

The dV numbers shown in red on the map are insertion burns which don't exactly match a specific Hohmann maneuver. For the ascent burns from Earth, I have factored in the Earth's equatorial velocity as well as gravity drag and atmospheric drag; these are all simplified best-case-scenario coplanar ascents. If you have a higher-latitude launch site or an inclination change then you'll need to increase your launch dV accordingly, but this can be done by simply adding the additional dV to the ascent dV shown. Same for on-orbit inclination changes; these can be calculated separately and tacked on. You should always do inclination changes at the highest apogee possible for maximum efficiency. For cislunar transfers, inclination changes at EML-1 are pretty miniscule.

So if you want to launch from Cape Canaveral to a 100x100km parking orbit and then do a GTO transfer, you'll want to increase your dV to make up for your latitude change and inclination change appropriately. But someone could create a separate reference table showing the additional dV required for each launch site and each destination.

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1 minute ago, sevenperforce said:

Yeah, the graphic is very large. You can download the pdf version here.

The dV numbers shown in red on the map are insertion burns which don't exactly match a specific Hohmann maneuver. For the ascent burns from Earth, I have factored in the Earth's equatorial velocity as well as gravity drag and atmospheric drag; these are all simplified best-case-scenario coplanar ascents. If you have a higher-latitude launch site or an inclination change then you'll need to increase your launch dV accordingly, but this can be done by simply adding the additional dV to the ascent dV shown. Same for on-orbit inclination changes; these can be calculated separately and tacked on. You should always do inclination changes at the highest apogee possible for maximum efficiency. For cislunar transfers, inclination changes at EML-1 are pretty miniscule.

So if you want to launch from Cape Canaveral to a 100x100km parking orbit and then do a GTO transfer, you'll want to increase your dV to make up for your latitude change and inclination change appropriately. But someone could create a separate reference table showing the additional dV required for each launch site and each destination.

best case simplification doesn't work, because we assume an equatorial sea-level launch, this create a Hohmann burn off a highly eccentric ellipse at its apoapsis running tangential to the surface of the earth, an impossible launch because of atmosphere and because horizontal acceleration at this starting velocity cannot be idealized as Hohmann. The launch phase has to assume dV placed into vertical velocity to create an ascent rate, that provides a temporal window for a horizontal (or off horizontal) burn to orbit. You can only extrapolate back to a Hohman transfer elliptical after you have developed a good positive verticle velocity and well into a turn. In this case you do not have idealized dV but are burning along a vector (generally with some vertical component) that is not at the Hohmann peri or apo.  If you launched sideways off of mount Everest along a Hohmann transfer you still crash into the earth around Burma, the transfer assumes that the burn aspect dtheta is trivial, in space you have the liberty of kicking several times at periapsis, but on earth you don't. The vertical aspect is a complete waste other than the energy of bring you up to a certain height which can be defined as MGH, only the horizontal burn dV is conserved. Then you have to factor in coefficient of drag, sub-mach-mach conversion dynamics at altitude,

Situation 1. Launch phase.

Ideal, very low coefficient of drag
Ideal, Infinite acceleration
Ideal, no Mach pressure effects.

Launch altitude = 100km, launch velocity = 1400 m/s ascent time = 142 seconds

Situation 2. Circulization phase 

Altitude 100,000, dV 7825

Combined = 9225.

Reality, Limit below 10,000 m Mach 0.95 Maximum acceleration is 15m/s rising to 18 m/s
Horizontal acceleration might be 2 m/s , drag losses might be 10% 55 seconds with a velocity of say 300 m/s
During the 55 seconds 907 dV is produced, however only 442 is conserved and 446 is lost (possibly more do to Mach effects).

Then you move to the next 10,000 m segment until your target altitude is reached. This could be at a separation stage, but here again its not an idealized hohmann because it still has vertical velocity and the idealized Hohmann begins at the periapsis.

The surface launch consideration should not be given at all in the graph, Instead a minimum (but unstable) orbit is at 80,000 meters.

Instead what you need is a object (like a Vbasic worksheet function) that uses CoD, Starting accleration, Maximum acceleration, starting surface horizontal velocity (latitude function), altitude, radius of earth at latitude. Makes some assumptions about the turn to horizontal, enter these is cells to the left of the function cell and then calculate dV. The best I think you can do is to assume something like at 30,000 m peri on a Hohmann transfer, but even there why bother. Next row should be the |normal| burn for inclination equitorial and a total dV.

 

Honestly if you are worried about 47dV at the hill radius but don't do some kind of homework on the launch then higher alt parts of the tables are meaningless.

 

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51 minutes ago, PB666 said:

best case simplification doesn't work, because we assume an equatorial sea-level launch, this create a Hohmann burn off a highly eccentric ellipse at its apoapsis running tangential to the surface of the earth, an impossible launch because of atmosphere and because horizontal acceleration at this starting velocity cannot be idealized as Hohmann. The launch phase has to assume dV placed into vertical velocity to create an ascent rate, that provides a temporal window for a horizontal (or off horizontal) burn to orbit. You can only extrapolate back to a Hohman transfer elliptical after you have developed a good positive verticle velocity and well into a turn.

Honestly if you are worried about 47dV at the hill radius but don't do some kind of homework on the launch then higher alt parts of the tables are meaningless.

Two points. First of all, the numbers for transfers between various orbits are still entirely valid even if you don't use the numbers for launch. For example, if you're staging for a cislunar mission at the space station, then you can simply use your launch vehicle's stated ISS payload and start from there. Moreover, because the periapse and apoapse burns are shown separately, you can start or end halfway through a Hohmann transfer. If your launch vehicle can deliver 10 tonnes to GTO then you can still calculate the required GEO circularization burn without necessarily knowing the exact dV burned to get onto the GTO; if you're coming back from a cislunar mission and passing through EML-1 then you'll be able to estimate the apogee burn for aerobraking without worrying about the launch dV.

Second, and more to the point, the surface periapse velocities given in the tables are decidedly not the same as the ones provided in the dV map.

Rather than going through the excessive pain of adjusting for T/W ratio and atmospheric drag and calculating the ideal gravity turn and everything else, I've simply used an average figure for total gravity drag and atmospheric drag in launches -- 1.5 km/s -- and added that on to the putative Hohmann transfer perigee velocity when determining the values for the dV map. I also subtracted the 465 m/s rotational speed of the Earth at the equator for the best-case simplification. Thus, even though a launch vehicle obviously does not make an instantaneous horizontal burn equal to the Hohmann transfer perigee velocity at the launch pad, it will be as if it burned that amount of dV plus an additional 1.5 km/s in order to get onto a trajectory that can be circularized. That's what is shown on the dV map.

Naturally, the launch dV requirements will be highly dependent on the launch vehicle; a narrow rocket with a high T/W ratio will be able to get away with lower launch dV than a wide rocket or a VTHL spaceplane with poorer thrust. But the launch dV values provided give a reasonably good estimate for the dV you need. In most cases, serious mission planning will involve the use of actual quoted figures for a given launch system, so you know what can be delivered to your initial parking orbit and you can plan from that starting point.

The launch dV values are also useful for comparative reasons. If your launch vehicle can deliver a given payload a 100x100km LEO with 1200 m/s of remaining dV, then you can predict the remaining dV for a launch to the ISS by looking at the difference between the circularization dVs and the launch dVs. This works with both the table and the map. Launching from the surface directly to the ISS will cost you 88 m/s more in your initial burn and 86 m/s more in your circular burn than launching to 100 km would. This may or may not be more efficient than first launching to 100x100 km and then doing a Hohmann transfer up, depending on inclination changes. However, if it's something like a launch to EML-1, then you will have significant savings by launching directly from the surface compared to launching to 100x100 km and then doing a transfer. This will limit your launch window, though.

All of these factors need to be considered. This just presents a common starting point.

Edited by sevenperforce
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8 hours ago, sevenperforce said:

The colors are pink, purple, and pale blue; depending on your computer settings the pale blue may appear to have a greenish tinge. 

The italized axis labels represent varying apsides. When the apsis on the horizontal axis matches the apsis on the vertical axis, the orbit is circular and has only one speed; when the axes don't match, there is a high apsis (the apoapse/apogee) and a low apsis (the periapse/perigee). Because the axes have the same labels, each possible combination appears twice, once in the pink region and once in the blue region. But since this is now an elliptical orbit without constant speed, the minimum speed (at apoapse) is shown in blue while the maximum speed (at periapse) is shown in pink.

For most coplanar orbits, the most efficient way to transfer between circular orbits at different altitudes (for example, to go from a 100 km circular parking orbit to the 400 km circular orbit of the ISS) is to enter an elliptical orbit with a perigee at 100 km and an apogee at 400 km, ride it up to its highest point, then transfer into a circular 400 km orbit with a circularization burn. Virtually all orbital maneuvers involve some sort of elliptic transfer. This table allows you to figure out how much speed you need to enter or to leave any of the given orbits. 

That's the thing; you don't necessarily have the velocity at a given apsis. If I'm in a 185x185 km circular parking orbit and want to send a satellite to geostationary orbit, I won't know how large of a burn I need. But glancing at the table, I see that the perigee speed for a 185km x 36,786km elliptical orbit is 10,268 m/s. My parking orbit speed is 7,793 m/s (shown in purple). So I need 3,275 m/s of dV to enter the geostationary transfer orbit.

Once the satellite completes the transfer, I can look at the apogee speed for that same 185km x 36,786km orbit to figure out how much additional velocity the satellite will need in order to circularize to a 36,786x36,786 km geostationary orbit.

So it's not just about finding perigee speed for a given apogee (or vice versa); it's about calculating the energy requirements for every different transfer or maneuver. 

The dV map I linked to shows all this even more clearly; it was just too large an image to embed here. 

You can actually define orbits by their angular momentums. 

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1 hour ago, sevenperforce said:

Oh, I have no doubt. But for executing Hohmann transfers, you need to know periapse and apoapse velocities so that you know what delta-v to apply. 

And you can know that. If you know one apsis and it's velocity then the angular momentum equation will tell you the other apsis' velocity (if you know its distance).

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1 hour ago, Bill Phil said:

And you can know that. If you know one apsis and it's velocity then the angular momentum equation will tell you the other apsis' velocity (if you know its distance).

You're missing the point: you don't know the velocity at the first apsis. 

If you need to do a Hohmann transfer from one circular orbit to another circular orbit, all you have is the distances. You don't have velocities at all. 

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easy v = sqrt(mu/r) and if you have velocity and r you have total momentum. But this isan issue of taste your method is valid you simply gloss over the kargest source of dV variation. 

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32 minutes ago, PB666 said:

easy v = sqrt(mu/r) and if you have velocity and r you have total momentum. But this isan issue of taste your method is valid you simply gloss over the kargest source of dV variation. 

v = sqrt(mu/r) gives you circular velocity, sure. How is that going to help you find apsis speed for an elliptical orbit with one apsis at r? It doesn't. 

Finding the periapse and apoapse velocities of an elliptical orbit based only on the periapse and apoapse distances requires you to solve for specific orbital energy using the semimajor axis, add the negative gravitational potential for the point you're interested in, and then convert specific kinetic energy into speed. Only then will you know how much delta v a given maneuver will consume. These tables provide the velocity for you, and the dV map takes it a step further by subtracting the circular velocities appropriately to show transfer dV at each point. 

If you think there's a simpler way to do it, I'm all ears. Tell me how you would calculate dV to get from a 185x185km circular equatorial parking orbit onto GTO, and how much additional dV it would take to get from GTO to GEO.

And sure, there will be variation in launch dV. That's why people would typically use quoted launch system payload capacity and start from LEO. But quoting estimated launch dV allows for comparative adjustment, so that's still useful while in no way detracting from the rest of the dV map. 

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10 hours ago, sevenperforce said:

You're missing the point: you don't know the velocity at the first apsis. 

If you need to do a Hohmann transfer from one circular orbit to another circular orbit, all you have is the distances. You don't have velocities at all. 

You solve the vis-viva equation. Or, you just measure it.

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1 hour ago, Bill Phil said:
11 hours ago, sevenperforce said:

You're missing the point: you don't know the velocity at the first apsis. 

If you need to do a Hohmann transfer from one circular orbit to another circular orbit, all you have is the distances. You don't have velocities at all. 

You solve the vis-viva equation. Or, you just measure it.

The table is the set of solutions to the vis-visa equations for the orbits provided.

How would you measure it? Like, use a simulator or something?

Edited by sevenperforce
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1 hour ago, sevenperforce said:

v = sqrt(mu/r) gives you circular velocity, sure. How is that going to help you find apsis speed for an elliptical orbit with one apsis at r? It doesn't. 

Finding the periapse and apoapse velocities of an elliptical orbit based only on the periapse and apoapse distances requires you to solve for specific orbital energy using the semimajor axis, add the negative gravitational potential for the point you're interested in, and then convert specific kinetic energy into speed. Only then will you know how much delta v a given maneuver will consume. These tables provide the velocity for you, and the dV map takes it a step further by subtracting the circular velocities appropriately to show transfer dV at each point. 

If you think there's a simpler way to do it, I'm all ears. Tell me how you would calculate dV to get from a 185x185km circular equatorial parking orbit onto GTO, and how much additional dV it would take to get from GTO to GEO.

And sure, there will be variation in launch dV. That's why people would typically use quoted launch system payload capacity and start from LEO. But quoting estimated launch dV allows for comparative adjustment, so that's still useful while in no way detracting from the rest of the dV map. 

Thats not exactly true because you know the semi-major axis of a circular orbit and for an elliptical orbit its

pe/2 + apo/2 + rsurface,central body and if you cannot calculate v at apo and peri, well need to spend more time on the wiki pages.

Did I not say this was just different ways of solving the problem. When you talk about orbits for me its potential and kinetic energy thats they way I think (IOW before KSP this is how I solved), so if I am not offered any solution, I will naturally solve for those two for any orbit I want to solve. If you take the integral of uH you can solve for potential energy for any two positions about a warp in space-time, so thats damn easy, at least for me. The rest is just velocity energy differentials. The reality is if you know how to calculated the celestial gravitational constant for any body, the math suddenly becomes much easier. 3.98E14.

See, in the game your conics show you the dV required for each segment, but for planning you only need to know total dV for each transfer. Let the game split them. If you had an excel table you simply put in your orbit your target orbit and the eccentrity and let the hidden vbasic function spit out a total dV, its pretty philosophical since you still haven't handled inclinations.

Theoretically for any switch you hohmann burn at the asc or des node (with a vector for inclination and at apo you burn both simultaneously for remaining inclination and circularization, but if you are catching target you might burn part of the orbit inorder to create an orbit that has an apo that coincided with the target and perform a second burn for an efficient use of dV for both inclination and orbit matching. In the game I do this alot for docking from launch, as part of my game I have a factory at 600k that needs to be resupplied, so I am going to use several burns, first one at peri to reach apo, then a second at apo to creat an orbit that will intercept target and finally a third burn to match target speed. I can throw inclination burns needed into this at my convenience. Doesnt really fit into the tabular form you provide. But what would be convenient is to know the total dV of the target intercept. The problem is that the greatest source of variation is in the launch to dV. If I let McJeb do it, its often a wash, or inefficient, if you do it manually you are often left short or excess of fuel.

So what I really need is a launch program that looks at things like CoD, the maximum dynamic pressure, maximum thrust to weight ratios,

 

Therefore we run into inaccuracies when you start dealing with things like 3-body problems, inclinations, interception, ect.

The highest energy inclination differences per degree however are the inclination changes required for circulation after launch, which you basically swept under the rug.
 

 

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4 minutes ago, PB666 said:

Thats not exactly true because you know the semi-major axis of a circular orbit and for an elliptical orbit its

pe/2 + apo/2 + rsurface,central body and if you cannot calculate v at apo and peri, well need to spend more time on the wiki pages.

We run into inaccuracies when you start dealing with things like 3-body problems, inclinations, interception, ect.

The highest energy inclination differences per degree however are the inclination changes required for circulation after launch, which you basically swept under the rug.

Wasn't really intending this particular dV map for use in the game; that's why I said it was for planning lunar missions. I could make the table for Kerbin easily enough.

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23 hours ago, sevenperforce said:

Wasn't really intending this particular dV map for use in the game; that's why I said it was for planning lunar missions. I could make the table for Kerbin easily enough.

been trying to get a cogent reply to the previous one but the GUI interface has been futzing up all the character mods, whenever I cut an past info here it completely messes up the formating. This is going to be low tech, sorry if it looks a bunch unreadable, blame it on the website. u is equal to the celestials gravitational constant, from this point we care nothing about gravity other than this constant, static gravity has meaning but not in the calculation of enery.

for any orbit above the surface of a celestial there is an a such that = pe/2 + apo/2 + rsurface,celestial for any a there is an r such that r = a on all points. For that r there is a potential energy from outside the hill sphere (r = infinity) to ra in which the potential energy difference is equal to u/r (very simple, but in real life its actually a negative value). Except close to the hill radius and/or L1 and L2, the potential energy of a static object is always double that of an orbiting object and also double its kinetic energy relative to the point-center mass.  So lets say that I have an orbit of 6370000 meters at v = 0. My exit potential is 62574638/kg No problem, if I do not add dv I will have an orbit about a point center that is basically two infinitely close lines that touch the center of the point mass (essentially it would be a black hole and I would fall into it). To create an orbit I need to add half the energy so I take (62574638/2) that give Ehalf  then (2E)0.5  = V = 7910 now to exit the orbit I would have to add another Ehalf of velocity. Now I already have a pretty good idea, because GSO is relatively close in energy to the hills radius Epot and So from the surface that is simply (2*u/r)0.5 = 11,187 so I kind of know that dV is going to be less than 3277.

So now I want to have an orbit that goes from Surface to GSO. Ok lets assume that both orbits are circular orbits.  To get to GSO I need to add 53121104/kg joules from an object just standing on a non-moving surface. The problem is that velocity gets me to GSO from the surface but it doesn't keep me there, I need to add additional energy to keep me there, so I have to add some more E to keep me. To stay at that distance all I need to do is add half the potential from GSO to exit this gives a velocity of 10756. Since I have already put 7910 into getting to my surface orbit I simply [cough] need dV 2345 which needs to be parse into an initiation burn and a completion burn. Its not going to be this good, there is waste because there are two burns required to reach a circle and two more required to reach GSO. Whereas if I launched strait up to GSO and burned it would be 13400 dV. Thus circularization is good at LEO and burn to GSO compared to burn up and circularization at GSO. Meaning if you have to burn vertical and horizontal, by all means do it at a lower altitude. However there are still better choices, because all I need to do is get a high enough vertical velocity to keep me from slamming into the thick atmosphere while I do a horizontal burn. I can get closer to the 10756 if I blow off what it took me to get to 10,000 meters (about 450 dV) and assume that I had a fast perfect burn to elliptical GSO from there, and avoiding the circularization.  There is a good space ecology reason for doing this and not circularizing first, any stages you burn through to reach GSO from a pe of say 10,000 to 80,000 meters, or even a bit into the circularization burn, they fall promptly back down to earth or into decay circularization to earth. 

The reality is the approach up to 80k alt is going to have to be an engineer approach, not pure math, a strategy of simulation based on maxima and NGO setpoints (e.g. maximum dynamic pressure) CoD, turns. But the point is at 80k you want to be on that GSO ellipse and very close to have accelerated to intercept velocity, and not spent much more energy fighting gravity than spent fighting drag (that is a reason for small craft getting CoD to absolute minimum is essential). At that point you can delta the difference between Erequired - EIdeal + Esurface velocity of earth at whatever latitude and determine the launch anomaly. Someone had a post the other day about Cygnus futzing up the parameters, but what they don't see is that drag is an estimate, the atmosphere is not perfectly predicable.

 

BTW GSO has a radius of 42,163.96 km = Roughly        CubeRoot( 398600442000000 /( 2*pi* 365.2422 /( 24* 60 * 60 * 364.2422 ))2)  ok maybe not roughly

Edited by PB666
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