Author Topic: Exhaustive dv map for lunar and cislunar missions  (Read 10508 times)

sevenperforce

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Exhaustive dv map for lunar and cislunar missions
« on: 03/30/2016 08:57 pm »
During my quest to come up with a viable way for SpaceX to service a manned lunar mission using their current platforms, I grew tired of all the laborious calculations I was running. So I did more calculations, and came up with this.

To start with, here are two reference tables showing the parameters for a variety of useful orbits around the Earth and the Moon.

Each combination of periapse and apoapse is provided. Circular orbits, where periapse and apoapse are equal, are highlighted in purple; all other orbits combining apsides show the periapse velocity in pink and the apoapse velocity in blue. All orbits are assumed to be coplanar.

This enables reasonably simple calculation of required dV for Hohmann transfer between various orbits. If you are at 100x100 km and want to move up to the ISS at 400 km, then you subtract the Hohmann transfer perigee velocity (6778x6478 = 7932) from your current velocity (7844) to get 88 m/s. That is your perigee burn. Then, once you swing up to 400 km, your velocity will have dropped to the Hohmann transfer apogee velocity (6478x6778=7581) and you will need to burn 87 m/s to get up to the 400x400 km speed of 7668. The same can be done in reverse for moving from a high orbit to a low orbit.

This also allows back-calculation of payload. If a launch provider quotes a particular payload capacity to GTO, starting at a particular parking orbit, then you can determine the size of the dV burn required to get onto the transfer orbit and thus find the payload capacity to the parking orbit.

I included the surface transfer "perigee velocity" but greyed out the perigee numbers, since they will need about 1.5-2 km/s added in order to account for gravity drag. This can be reduced for prograde orbits but will need to be increased for retrograde orbits. The perilune velocities for transfer from the lunar surface also need gravity drag added...it is usually around 139 m/s, if my estimate is correct.

Burning to Lagrange points 1 and 2 is a little different. The perigee velocity is the same, but for the target apogee velocity, you need to match the 362,600x362,600 km "Luna" circular velocity. This means the apogee burn is lower than it would be for EML-1 and higher than it would be for EML-2. For EML-4 and EML-5 you can just use the "Luna" target orbit. This assumes that the moon is at perigee itself. On the flip side, because EML-1 and EML-2 are stationary from the perspective of the moon, you need only to cancel your apogee speed with a retrograde burn, not match speed.

The Hill radius is included as the highest possible orbit for bi-elliptic transfers.

Lunar slingshot is the free return trajectory used by Apollo; it also allows inexpensive transfer to EML-2, though this is not shown in the tables.

All this was pretty cool. But I decided to do more.

This thumbnail image (also in a PDF attachment) links to a dV map showing all possible transfers of interest between the Earth and the moon. Periapse and apoapse burns are color-coded. dV shown in red is an injection or ascent burn which doesn't exactly match any Hohmann transfer for one reason or another. Gravity drag and atmospheric drag are provided as needed. The slingshot orbit will loop into free-return if you don't execute any further burn past the initial injection; however, it allows inexpensive transfer to EML-2 as well.

Obviously, it is not to scale. Sorry about that...wasn't really any way I was going to pull a scale image off. It is also obviously a patched conic approach. I'm pretty sure the numbers are correct, but it's hard to double-check this stuff (since I can't very well fly a mission and see), so if you have corrections please let me know.

If you want to combine multiple transfers (e.g. for a bi-elliptic transfer or for a slide across a lagrange point) then you can skip the apogee burn and simply match velocities with your target transfer's apogee velocity, but you may need to refer to the tables to make those numbers work.

Hope this is useful!
« Last Edit: 03/30/2016 09:03 pm by sevenperforce »

redliox

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Re: Exhaustive dv map for lunar and cislunar missions
« Reply #1 on: 03/30/2016 09:11 pm »
300 kilometers doesn't sound particular high for a high orbit.  What about things such as Distant Retrograde Orbit or the 'Frozen' elliptical orbits?  Those seem more likely first choices NASA would pick for long-term stability although Low Lunar Orbit would always be a precursor to landing.
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sevenperforce

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Re: Exhaustive dv map for lunar and cislunar missions
« Reply #2 on: 03/30/2016 09:30 pm »
I picked 310 km to match the initial insertion used by Apollo, though I suppose it's not actually a high orbit.

The frozen orbits are inclination changes and I wanted to keep it a bit simpler than that.

TrevorMonty

Re: Exhaustive dv map for lunar and cislunar missions
« Reply #3 on: 03/31/2016 12:16 am »
Great work Seven perforce.

sevenperforce

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Re: Exhaustive dv map for lunar and cislunar missions
« Reply #4 on: 03/31/2016 02:38 pm »
Great work Seven perforce.
Thanks.

I made an adjustment -- I was using lunar-orbital-velocity matching for EML-1 and EML-2 from Earthside, but I should have been using lunar-period matching. Corrected here:

And here's the corrected dV map:

Steven Pietrobon

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Re: Exhaustive dv map for lunar and cislunar missions
« Reply #5 on: 04/01/2016 06:16 am »
Great graph. Could you also add the direct delta-V's from LEO to LLO, instead of having to pass through the EML-1 and EML-2 tollbooths? The 100 km LEO orbit is not used by anyone to get to anywhere as it is too unstable. I think you can delete that from your graph. The descriptions for the orbits seems to be quite far from their respective marks. Could you move the descriptions next to their marks? The Earth Hill Radius Could be replaced with Mars, as that is a more useful destination.
Akin's Laws of Spacecraft Design #1:  Engineering is done with numbers.  Analysis without numbers is only an opinion.

sevenperforce

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Re: Exhaustive dv map for lunar and cislunar missions
« Reply #6 on: 04/01/2016 02:42 pm »
Great graph. Could you also add the direct delta-V's from LEO to LLO, instead of having to pass through the EML-1 and EML-2 tollbooths? The 100 km LEO orbit is not used by anyone to get to anywhere as it is too unstable. I think you can delete that from your graph. The descriptions for the orbits seems to be quite far from their respective marks. Could you move the descriptions next to their marks? The Earth Hill Radius Could be replaced with Mars, as that is a more useful destination.
I included the 100 km circular orbit for a few reasons. First of all, launch providers quoting a payload to LEO are going to invariably quote the maximum possible payload they could loft into any orbit, so they will typically use a 100x100km orbit as a standard reference. Second, this altitude isn't used for extended parking, but it can be used as a precursor to a transfer when you want to change the longitude of your Hohmann transfer semimajor axis. It is also useful for planning re-entry; the apogee burn to lower your perigee to 100 km is roughly what you need to ensure proper re-entry angle, so you want 100 km as a reference from any orbit.

Computing dV directly from LEO to LLO would be a solution to the three-body problem, and would depend quite heavily on the route you wanted to take. I'm also pretty sure that such an orbit would be more expensive from a dV perspective than going through either EML1 or EML2.

I kept the orbit labels separate from the nodes to avoid clutter, but I could move them a bit closer.

Giving Mars as the highest destination would be a four-body problem, which is even worse than a three-body problem. I provided the Hill radius not as a destination, but as an apogee for lowest-possible-dV bi-elliptic transfers.

Steven Pietrobon

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Re: Exhaustive dv map for lunar and cislunar missions
« Reply #7 on: 04/02/2016 04:19 am »
First of all, launch providers quoting a payload to LEO are going to invariably quote the maximum possible payload they could loft into any orbit, so they will typically use a 100x100km orbit as a standard reference.

Atlas V users guide is 200 km minimum. Delta IV users guide is 185 km. Couldn't find a graph in the Ariane 5 guide, but the minimum to ISS is 200 km. Proton guide is 180 km.

Quote
Second, this altitude isn't used for extended parking, but it can be used as a precursor to a transfer when you want to change the longitude of your Hohmann transfer semimajor axis.

I've never seen a launch that goes that low before transfer. Its normally around 200 km.

Quote
It is also useful for planning re-entry; the apogee burn to lower your perigee to 100 km is roughly what you need to ensure proper re-entry angle, so you want 100 km as a reference from any orbit.

100 km is too high. You need to target a perigee of around 37 km to ensure re-entry.

Quote
Computing dV directly from LEO to LLO would be a solution to the three-body problem, and would depend quite heavily on the route you wanted to take.

You don't need to go to the complexity of a three-body system. Just target apogee at Lunar distance and use the hyperbolic transfer formula to perform insertion. Vhyp² = Vesc² + Vinf² where Vinf is your speed relative to the Moon at apogee (assuming the Moon is not there), Vesc is the escape speed from the Moon (Vesc = sqrt(2)*Vcirc, where Vcirc is your required orbital speed around the Moon) and Vhyp is your speed near the Moon before insertion. Subtract Vcirc from Vhyp to calculate your delta-V.

Quote
I'm also pretty sure that such an orbit would be more expensive from a dV perspective than going through either EML1 or EML2.

I'm pretty sure that is wrong. Going direct using Hohman transfers will always have a lesser delta-V. Here are some numbers using NASA's ground rules and assumptions (information provided by Ross Tierney).

TLI: 3175 m/s
LOI: 1018 m/s
Total: 4193 m/s

EML-1: 3125 m/s
EML-1 Insertion: 640 m/s
EML-1 to LLO: 1333 m/s
Total: 5098 m/s

EML-2: 3225 m/s
Perilune:  184 m/s
EML-2 Insertion: 147 m/s
EML-2 to LLO: 1333 m/s
Total: 4889 m/s

Quote
Giving Mars as the highest destination would be a four-body problem, which is even worse than a three-body problem.

Four body? Its three-body (Sun, Earth and Spacecraft) as Mars gravity is insignificant near Earth and vice versa. Again, you also don't need to go to that detail. Just pick the greatest Mars radius around the Sun and use the hyperbolic transfer formula. Its so simple, a child can work this out, as shown in this colouring book.

http://www.amazon.com/Conic-Sections-Celestial-Mechanics-Coloring/dp/1936037106
« Last Edit: 04/02/2016 04:33 am by Steven Pietrobon »
Akin's Laws of Spacecraft Design #1:  Engineering is done with numbers.  Analysis without numbers is only an opinion.

sevenperforce

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Re: Exhaustive dv map for lunar and cislunar missions
« Reply #8 on: 04/02/2016 01:52 pm »
The Martian transfer is a four-body problem in that you have to account for the relative positions of the two bodies, but sure.

Do you know what orbital altitudes those NASA reference figures were calculated for? I'm getting much lower figures for the Lagrange transfers from 185km LEO to 310 km LO...4,305 m/s for the EML-1 transfer and 3,898 m/s for the EML-2 transfer.

the_other_Doug

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Re: Exhaustive dv map for lunar and cislunar missions
« Reply #9 on: 04/02/2016 03:26 pm »
Quote
Second, this altitude isn't used for extended parking, but it can be used as a precursor to a transfer when you want to change the longitude of your Hohmann transfer semimajor axis.

I've never seen a launch that goes that low before transfer. Its normally around 200 km.

Really?  Apollo used to insert into a short-term LEO parking orbits of between 100 and 110 statute miles, circular.  (It was higher for earlier flights;  the J missions, to keep as much delta-V for TLI as possible, inserted into the lowest possible orbit, usually targeted for about 98 statute miles circular.)  This amounts to, what, around 130 km?

Definitely lower than 200 km...
-Doug  (With my shield, not yet upon it)

Proponent

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Re: Exhaustive dv map for lunar and cislunar missions
« Reply #10 on: 04/02/2016 03:51 pm »
Are you sure that was 98 statute miles?  That seems really low.  My bet is 98 nautical miles, i.e., 176 km.

the_other_Doug

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Re: Exhaustive dv map for lunar and cislunar missions
« Reply #11 on: 04/02/2016 06:27 pm »
Are you sure that was 98 statute miles?  That seems really low.  My bet is 98 nautical miles, i.e., 176 km.

Earlier Apollos launched into 180 to 190 km circular parking orbits, or just more than 100 nautical miles.  Apollo 17's parking orbit (representative of the J missions), per Encyclopedia Astronautica, was 168 by 170 km. That's 104 by 105 statute miles, or 90 by 91 nautical miles.  So, yeah, by memory I was off by about 5 statute miles on the later Apollo parking orbits.

Even for the earlier Apollos, though, the parking orbit was lower than 200 km by between 10 and 20 km.  The later ones were 30 km lower.
-Doug  (With my shield, not yet upon it)

Steven Pietrobon

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Re: Exhaustive dv map for lunar and cislunar missions
« Reply #12 on: 04/03/2016 06:21 am »
Apollo used to insert into a short-term LEO parking orbits of between 100 and 110 statute miles, circular.  (It was higher for earlier flights;  the J missions, to keep as much delta-V for TLI as possible, inserted into the lowest possible orbit, usually targeted for about 98 statute miles circular.)  This amounts to, what, around 130 km?

Definitely lower than 200 km...

I said "around 200 km", which implies a range of 150 to 250 km if rounding to one significant figure. I certainly did not mean three figure accuracy! Sorry for the confusion. Apollo's 8 to 14 were close to 100 nautical miles, which is 185 km. The J missions were close to 90 nautical miles or 167 km (I did not know that before). Still, that is closer to 200 km than 100 km, which was the point I was trying to argue.
« Last Edit: 04/03/2016 06:29 am by Steven Pietrobon »
Akin's Laws of Spacecraft Design #1:  Engineering is done with numbers.  Analysis without numbers is only an opinion.

TrevorMonty

Re: Exhaustive dv map for lunar and cislunar missions
« Reply #13 on: 04/03/2016 12:12 pm »
First of all, launch providers quoting a payload to LEO are going to invariably quote the maximum possible payload they could loft into any orbit, so they will typically use a 100x100km orbit as a standard reference.

Atlas V users guide is 200 km minimum. Delta IV users guide is 185 km. Couldn't find a graph in the Ariane 5 guide, but the minimum to ISS is 200 km. Proton guide is 180 km.

Quote
Second, this altitude isn't used for extended parking, but it can be used as a precursor to a transfer when you want to change the longitude of your Hohmann transfer semimajor axis.

I've never seen a launch that goes that low before transfer. Its normally around 200 km.

Quote
It is also useful for planning re-entry; the apogee burn to lower your perigee to 100 km is roughly what you need to ensure proper re-entry angle, so you want 100 km as a reference from any orbit.

100 km is too high. You need to target a perigee of around 37 km to ensure re-entry.

Quote
Computing dV directly from LEO to LLO would be a solution to the three-body problem, and would depend quite heavily on the route you wanted to take.

You don't need to go to the complexity of a three-body system. Just target apogee at Lunar distance and use the hyperbolic transfer formula to perform insertion. Vhyp² = Vesc² + Vinf² where Vinf is your speed relative to the Moon at apogee (assuming the Moon is not there), Vesc is the escape speed from the Moon (Vesc = sqrt(2)*Vcirc, where Vcirc is your required orbital speed around the Moon) and Vhyp is your speed near the Moon before insertion. Subtract Vcirc from Vhyp to calculate your delta-V.

Quote
I'm also pretty sure that such an orbit would be more expensive from a dV perspective than going through either EML1 or EML2.

I'm pretty sure that is wrong. Going direct using Hohman transfers will always have a lesser delta-V. Here are some numbers using NASA's ground rules and assumptions (information provided by Ross Tierney).

TLI: 3175 m/s
LOI: 1018 m/s
Total: 4193 m/s

EML-1: 3125 m/s
EML-1 Insertion: 640 m/s
EML-1 to LLO: 1333 m/s
Total: 5098 m/s

EML-2: 3225 m/s
Perilune:  184 m/s
EML-2 Insertion: 147 m/s
EML-2 to LLO: 1333 m/s
Total: 4889 m/s

Quote
Giving Mars as the highest destination would be a four-body problem, which is even worse than a three-body problem.

Four body? Its three-body (Sun, Earth and Spacecraft) as Mars gravity is insignificant near Earth and vice versa. Again, you also don't need to go to that detail. Just pick the greatest Mars radius around the Sun and use the hyperbolic transfer formula. Its so simple, a child can work this out, as shown in this colouring book.

http://www.amazon.com/Conic-Sections-Celestial-Mechanics-Coloring/dp/1936037106

This LEO to EML1 DV (3765) is for fast transit ie a few days. There is a slower few weeks transit which is around 3100m/s.

Are you sure about EML1- LLO DV (1333)?. This would make it 3200m/s to surface, I thought it was around 2700m/s.
« Last Edit: 04/03/2016 12:17 pm by TrevorMonty »

Steven Pietrobon

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Re: Exhaustive dv map for lunar and cislunar missions
« Reply #14 on: 04/04/2016 10:17 am »
Are you sure about EML1- LLO DV (1333)?. This would make it 3200m/s to surface, I thought it was around 2700m/s.

That's the value given in the list of values that Ross gave. I repeat them below.

"kraisee wrote on 03/10/2009 03:23 PM

Okay, here are some rudimentary Ground Rules and Assumptions for anyone attempting to prepare architecture options the correct way.   These are based on NASA's current GR&A's extrapolated for the different architectures.

Boil-off assumptions:
LOX/LH2:   0.35% per day.
LOX/CH4:   0.20% per day.
LOX/RP1:   0.20% per day.
MMH/N2O4:   0.00% per day.
Max Loiter in LEO:   4 days.
Max TLI Transit Time:   3 days.
Max LLO Loiter Time:   3 days.

dV Budget assumptions:

Direct to LLO (3 days to LLO):
TLI:   3,175.00m/s + 1% = 3,206.75m/s (performed by MPS)
TCM:   2.00m/s + 1% = 2.02m/s (performed by RCS)
LOI to HLO (100x10,000km):   503.00m/s + 1% = 508.03m/s (performed by MPS)
Plane Change (90deg):   476.00m/s + 1% = 480.76m/s (performed by MPS)
Circularization to LLO (100x100km):   515.00m/s + 1% = 520.15m/s (performed by MPS)

Direct to EML-1 (4 days to LLO):
TLI to EML-1:   3,125m/s + 1% = 3,156.25m/s (performed by MPS)
TCM:   2.00m/s + 1% = 2.02m/s (performed by RCS)
EML-1 Insertion:   640m/s + 1% = 646.40m/s (performed by MPS)
EML-1 to LLO (100x100km):   1,333m/s + 1% = 1,346.33m/s (performed by MPS)

Swingby to EML-2 (6 days to LLO):
TLI:   3,225.00m/s + 1% = 3,257.25m/s (performed by MPS)
TCM:   2.00m/s + 1% = 2.02m/s (performed by RCS)
Perilune:   184.00m/s + 1% = 185.84 (performed by MPS)
EML-2 Insertion:   147.00m/s + 1% = 148.47m/s (performed by MPS)
EML-2 to LLO (100x100km):   1,333m/s + 1% = 1,346.33m/s (performed by MPS)

Descent from 100x100km:
LLO Attitude Control:   5.50m/s + 1% = 5.56m/s
DOI:   19.40m/s + 1% = 19.56m/s (performed by RCS)
Settling:   2.7m/s + 1% =2.73m/s (performed by RCS)
Descent RCS:   11.00m/s + 1% = 11.1m/s (performed by RCS)
Powered Descent:   2,030.00m/s + 1% = 2,050.30m/s (performed by MPS)
(For staged descent:   75% = 1,537.72m/s & 25% = 512.58m/s)

Engine performance:
J-2X:   448.2s Vac. Isp.   294,000lb Vac. Thrust.
RL-10A-4-2:   451.0s Vac. Isp.   22,300lb Vac. Thrust.
RL-10B-2:   465.5s* Vac. Isp.   24,750lb Vac. Thrust.
Current LSAM DS Engine (RL-10-derived):   448.6sec Vac. Isp.   18,650lb Vac. Thrust.
Current LSAM DS Engine (AJ10-derived):   320.0s Vac. Isp.   5,500lb Vac. Thrust.
Current Orion SM Engine (AJ10-derived):   326.0s Vac. Isp.   9,500lb Vac. Thrust.

* Note I am attempting to re-validate this Isp figure.   Possibly to be amended down to ~462s."
Akin's Laws of Spacecraft Design #1:  Engineering is done with numbers.  Analysis without numbers is only an opinion.

sevenperforce

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Re: Exhaustive dv map for lunar and cislunar missions
« Reply #15 on: 05/18/2016 05:07 pm »
Re-evaluating minimum dV to get to lunar orbit (at least the minimum for reasonably fast transfers; e.g., a week is fine but a month is not)....

Steven's quoted NASA GR&A gives 3,225 m/s as the dV for the TLI on a lunar swingby to EML-2, but Robert Farquhar gives 10,310 fps or  3,142.5 m/s as the TLI dV. Not sure what the significance of the difference is. Both Farquhar and the GR&A list 184 m/s as the perilune dV.

Swingby is at 60 nautical miles altitude, or roughly 110 km. The perilune burn of 184 m/s is presumably a retrograde burn, decreasing the aposelene from infinity to one that matches EML-2. A lunar orbit with periselene at 110 km and aposelene at EML-2 has a periselene velocity of 2,258 m/s; if you simply burn retrograde here, it should only take a total of 629 m/s more to circularize at 110 km, for a total dV budget of 3,955.5 m/s to LLO.

Alternatively, if you continue up to EML-2 (with that 184 m/s Oberth-assisted burn) then you can raise your periselene to a 310 km parking orbit with a mere 5 m/s puff of RCS. Once you swing back around, you'll end up needing 593 m/s to circularize at 310 km, for a total dV budget of 3,924.5 m/s. That's just shy of 1 km/s cheaper than the EML-2 swingby given in the GR&A.

fregate

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Re: Exhaustive dv map for lunar and cislunar missions
« Reply #16 on: 12/25/2017 10:30 am »
There are multiple NASA Ground Rules and Assumptions
I would like to add delta-V for Near Rectilinear Orbit (NRO) as a chosen destination of DSG in cis-Lunar Space
- TLI burn delta V 3,150 m/sec (from circular 200 km LEO parking orbit)
- Mid-course corrections delta V 75 m/sec
- Perilune burn delta V 178 m/sec
- NRO Insertion delta V 250.5 m/sec
- Rendezvous and docking delta V 100 m/sec
Subtotal inbound leg of the mission delta V 3,753.5 m/sec

Source: Options for Staging Orbits in Cis-Lunar Space
« Last Edit: 12/25/2017 10:33 am by fregate »
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redliox

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Re: Exhaustive dv map for lunar and cislunar missions
« Reply #17 on: 01/29/2018 09:55 pm »
There are multiple NASA Ground Rules and Assumptions
I would like to add delta-V for Near Rectilinear Orbit (NRO) as a chosen destination of DSG in cis-Lunar Space
- TLI burn delta V 3,150 m/sec (from circular 200 km LEO parking orbit)
- Mid-course corrections delta V 75 m/sec
- Perilune burn delta V 178 m/sec
- NRO Insertion delta V 250.5 m/sec
- Rendezvous and docking delta V 100 m/sec
Subtotal inbound leg of the mission delta V 3,753.5 m/sec

Source: Options for Staging Orbits in Cis-Lunar Space

Good numbers for reference!

Personally, I'm curious how easily a spacecraft, whether Orion or the proposed Deep Space Gateway, could move between NRO and the cislunar Lagrange points.  One of the proposed assets for a Gateway station is the ability to move around cislunar space.  EM-1 is useful for lunar assets and EM-2 as a Martian waypoint, and 4 and 5 could have their own uses.
"Let the trails lead where they may, I will follow."
-Tigatron

fregate

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Re: Exhaustive dv map for lunar and cislunar missions
« Reply #18 on: 12/09/2018 03:50 am »
Steven, it seems to be that Roscosmos in favor of selecting Russian Lunar Orbital station on a polar circular Lunar orbit with an altitude of 10,000 km. Could you please explain how to calculate ideal LOI Delta-V for such destination in cis-lunar space?
"Selene, the Moon. Selenginsk, an old town in Siberia: moon-rocket  town" Vladimir Nabokov

Steven Pietrobon

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Re: Exhaustive dv map for lunar and cislunar missions
« Reply #19 on: 12/09/2018 08:33 am »
See page four of my paper

S. S. Pietrobon, "Lunar orbit propellant transfer," 8th Int. Aerospace Congress, Adelaide, Australia, Sep. 1999.
http://www.sworld.com.au/steven/pub/IAC99pap.pdf

From Baikonur theta = 51.6 - 18.1 = 33.5°. Using (3) this gives v_MS = 938.7 m/s (relative speed at infinity of spacecraft to Moon). At h_M = 10,000 km, v_o = 646.3 m/s. Using (4) this gives the required delta-V as 663.9 m/s.

To go into a 100 km circular the apolune burn is v_o2 - v_a = 646.3 - 336.3 = 310.0 m/s. The perilune burn is v_p - v_o1 = 2147.7 - 1633.2 = 514.5 m/s, requiring a total delta-v of 824.5 m/s. However, if we go directly to 100 km circular, the delta-V is only 860 m/s! I wish NASA, Roscosmos, etc would just stop with the Gateway. It is just plain stupid!
« Last Edit: 12/09/2018 08:33 am by Steven Pietrobon »
Akin's Laws of Spacecraft Design #1:  Engineering is done with numbers.  Analysis without numbers is only an opinion.

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