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Earth-Moon orbit

Scientific discovery and discussion
GoSeigen
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Earth-Moon orbit

#291834

Postby GoSeigen » March 18th, 2020, 5:48 am

Does anyone know how feasible it would be to put a satellite into an Earth-Moon orbit, by which I mean the satellite would regularly pass by both the moon and the earth on each circuit at a low altitude. Such an orbit would be markedly elliptical I guess, but could the motion be arranged such that only minimal adjustment to the satellite's path is needed, similar to satellites in a low Earth orbit? What would the likely orbital period be? I guess there might be a variety of lower or higher orbits that would work?

If so, why has no-one done it yet?

GS

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Re: Earth-Moon orbit

#291840

Postby DeBriele » March 18th, 2020, 7:07 am

I suspect that for approaches at your 'low' altitudes, the dynamics of earth/moon gravity would result in a very unstable orbit, with a requirement for much course correction and hence fuel/power.

At some sufficient distance, I imagine the earth/moon gravitational effect could reduce close enough to a single point to allow a stable large radius orbit. If then very highly elliptical, to allow your close (and now very occasional!) approaches, then I imagine the earth/moon dynamics on each pass could still cause significant perturbation which would need correction. And also have no idea then whether such an elliptical orbit would get to the point of being upset by other bodies?

DB (not a physicist)

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Re: Earth-Moon orbit

#291843

Postby JohnB » March 18th, 2020, 7:18 am

You could set up an elliptical orbit with 200 mile perigee (closed point to Earth) and 250,000 mile apogee (furthest point), but the time taken to complete it would be fixed, and it would most unlikely the Moon would be at the right point in its orbit at apogee.

GoSeigen
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Re: Earth-Moon orbit

#291860

Postby GoSeigen » March 18th, 2020, 8:29 am

JohnB wrote:You could set up an elliptical orbit with 200 mile perigee (closed point to Earth) and 250,000 mile apogee (furthest point), but the time taken to complete it would be fixed, and it would most unlikely the Moon would be at the right point in its orbit at apogee.



I hear DeBriele's points on perturbation but isn't this is just a matter of careful calculation? The moon's gravity is only around 1% of the earth's anyway, and if the perturbation were the same on each pass by making the period exactly a multiple of the moon's, then the maths would be greatly simplified wouldn't it, and any adjustments to orbit fairly minimal? Have I missed anything?

And the next question was about how useful such a satellite might be. I'm guessing not very useful, because you might as well just put a satellite in moon orbit. But what if it were something like the ISS: would there be any advantage to passing by the moon on a regular basis?

I'm guessing the major benefit to putting an object in space is being in space itself. Whether that is near the earth or near the moon is a very minor consideration compared to the respective costs involved. Would that be why it hasn't been done?


GS

dspp
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Re: Earth-Moon orbit

#291871

Postby dspp » March 18th, 2020, 8:53 am

GoSeigen wrote:
JohnB wrote:You could set up an elliptical orbit with 200 mile perigee (closed point to Earth) and 250,000 mile apogee (furthest point), but the time taken to complete it would be fixed, and it would most unlikely the Moon would be at the right point in its orbit at apogee.



I hear DeBriele's points on perturbation but isn't this is just a matter of careful calculation? The moon's gravity is only around 1% of the earth's anyway, and if the perturbation were the same on each pass by making the period exactly a multiple of the moon's, then the maths would be greatly simplified wouldn't it, and any adjustments to orbit fairly minimal? Have I missed anything?

And the next question was about how useful such a satellite might be. I'm guessing not very useful, because you might as well just put a satellite in moon orbit. But what if it were something like the ISS: would there be any advantage to passing by the moon on a regular basis?

I'm guessing the major benefit to putting an object in space is being in space itself. Whether that is near the earth or near the moon is a very minor consideration compared to the respective costs involved. Would that be why it hasn't been done?


GS


The orbital period is set by (crudely) the altitude of the orbit (more precisely so as to keep KE + PE constant). I've put some relevant links below in case you fancy building a spreadsheet :)

Low Earth Orbit at 160km altitude has a period of 90-minutes.
Semi-synchronous Earth Orbit (SSO) at 20,200 km has a period of approximately 12 hours.
Geosynchronous Earth Orbit at 36,000km has a period of 24-hours.
Moon Orbit of Earth at 384,000km has a period of approx 28-days.

What you are seeking is a special case of a special case, i.e. Trans-Lunar Injection orbit with a free return trajectory, that then repeats with no further orbital injection burns (maybe just trim). So the orbit needs to have a period of approximately 28 days or a multiple thereof, and will of course be elliptical to fit these constraints. It might be either a simple egg shaped, or figure-8 shaped, but I think the geometricians would consider them both to be capable of being calculated as ellipses (I'm not sure). However this indeed is possible, with various monthly periods, and I will let wiki give you the explanation of the ones Schwaniger calculated (see my last link below for diagrams etc).

Using the simplified model where the orbit of the Moon around the Earth is circular, Schwaniger found that there exists a free-return trajectory in the plane of the orbit of the Moon which is periodic. After returning to low altitude above the Earth (the perigee radius is a parameter, typically 6555 km) the spacecraft would start over on the same trajectory. This periodic trajectory is counter-rotational (it goes from east to west when near the Earth). It has a period of about 650 hours (compare with a sidereal month, which is 655.7 hours, or 27.3 days). Considering the trajectory in an inertial (non-rotating) frame of reference, the perigee occurs directly under the Moon when the Moon is on one side of the Earth. Speed at perigee is about 10.91 km/s. After 3 days it reaches the Moon's orbit, but now more or less on the opposite side of the Earth from the Moon. After a few more days, the craft reaches its (first) apogee and begins to fall back toward the Earth, but as it approaches the Moon's orbit, the Moon arrives, and there is a gravitational interaction. The craft passes on the near side of the Moon at a radius of 2150 km (410 km above the surface) and is thrown back outwards, where it reaches a second apogee. It then falls back toward the Earth, goes around to the other side, and goes through another perigee close to where the first perigee had taken place. By this time the Moon has moved almost half an orbit and is again directly over the craft at perigee. Other cislunar trajectories are similar but do not end up in the same situation as at the beginning, so cannot repeat.[2] . There will of course be similar trajectories with periods of about two sidereal months, three sidereal months, and so on. In each case, the two apogees will be further and further away from Earth. These were not considered by Schwaniger.

I think out-of-plane orbits would be unstable and so not repeat.

https://en.wikipedia.org/wiki/Orbit
https://en.wikipedia.org/wiki/Geocentric_orbit
https://en.wikipedia.org/wiki/Trans-lunar_injection
https://en.wikipedia.org/wiki/Free-return_trajectory

Injection/transfer orbits have their uses, but also have limitations. For non-time-sensitive (dead) freight they tend to be very energy efficient but slow.

regards, dspp

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Re: Earth-Moon orbit

#291876

Postby UncleEbenezer » March 18th, 2020, 9:03 am

Your simplest case to visualise would be a (distant) orbit around the Earth/Moon's common centre of gravity (maybe call it a tidal orbit 8-) ). But I guess that's not quite what you had in mind?

The fundamental maths of this should be no more than quadratic. Complex, but no more so than any space mission beyond Earth's orbit. If you're wanting some figure-of-eight I wouldn't care to predict how near you could get, but I expect googling could find you software that'll visualise orbits for you..

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Re: Earth-Moon orbit

#291884

Postby dspp » March 18th, 2020, 9:15 am

UncleEbenezer wrote:Your simplest case to visualise would be a (distant) orbit around the Earth/Moon's common centre of gravity (maybe call it a tidal orbit 8-) ). But I guess that's not quite what you had in mind?

The fundamental maths of this should be no more than quadratic. Complex, but no more so than any space mission beyond Earth's orbit. If you're wanting some figure-of-eight I wouldn't care to predict how near you could get, but I expect googling could find you software that'll visualise orbits for you..


You could do it in Kerbal

https://en.wikipedia.org/wiki/Kerbal_Space_Program
https://wiki.kerbalspaceprogram.com/wiki/Main_Page

Enjoy :)

dspp

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Re: Earth-Moon orbit

#291954

Postby ReformedCharacter » March 18th, 2020, 11:51 am

GoSeigen wrote:
The moon's gravity is only around 1% of the earth's anyway.

GS

I don't think that's correct:

the Moon's gravity is 1/6th (16.7%) of the Earth's gravity.

https://www.moonconnection.com/moon_gravity.phtml

RC

GoSeigen
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Re: Earth-Moon orbit

#291958

Postby GoSeigen » March 18th, 2020, 12:02 pm

ReformedCharacter wrote:
GoSeigen wrote:
The moon's gravity is only around 1% of the earth's anyway.

GS

I don't think that's correct:

the Moon's gravity is 1/6th (16.7%) of the Earth's gravity.

https://www.moonconnection.com/moon_gravity.phtml

RC


Sorry, I was referring to the standard gravitational parameter GM which is also a function of the body's mass and affects the period of an object orbiting the body.

For Earth GM = 4E14 and for the Moon it's 5E12.

You're right about gravitational acceleration on the surface of each planet, but that is a different quantity altogether. [EDIT: Gravitational acceleration at an earth radius's distance from the centre of the moon is about 1% of earth's g on its own surface.]


GS

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Re: Earth-Moon orbit

#292054

Postby XFool » March 18th, 2020, 4:44 pm

GoSeigen wrote:Does anyone know how feasible it would be to put a satellite into an Earth-Moon orbit, by which I mean the satellite would regularly pass by both the moon and the earth on each circuit at a low altitude. Such an orbit would be markedly elliptical I guess, but could the motion be arranged such that only minimal adjustment to the satellite's path is needed, similar to satellites in a low Earth orbit? What would the likely orbital period be? I guess there might be a variety of lower or higher orbits that would work?

If so, why has no-one done it yet?

I'd say, in a way, they have already done it! They 'did it', for instance, with every one of the manned Lunar landings in the 1960s and 70s. OK, they 'braked' during the Lunar part and didn't actually complete on return to Earth. If Apollo 13 had not been so successful it might have ended up in such an orbit. But would the Moon always be 'there'? I'm not sure of that.

Period? Off hand I don't know, but if you take the time from Earth orbit to the Moon with Apollo, double it and add a bit you shouldn't be far off.

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Re: Earth-Moon orbit

#292076

Postby dspp » March 18th, 2020, 6:14 pm

XFool wrote:
GoSeigen wrote:Does anyone know how feasible it would be to put a satellite into an Earth-Moon orbit, by which I mean the satellite would regularly pass by both the moon and the earth on each circuit at a low altitude. Such an orbit would be markedly elliptical I guess, but could the motion be arranged such that only minimal adjustment to the satellite's path is needed, similar to satellites in a low Earth orbit? What would the likely orbital period be? I guess there might be a variety of lower or higher orbits that would work?

If so, why has no-one done it yet?

I'd say, in a way, they have already done it! They 'did it', for instance, with every one of the manned Lunar landings in the 1960s and 70s. OK, they 'braked' during the Lunar part and didn't actually complete on return to Earth. If Apollo 13 had not been so successful it might have ended up in such an orbit. But would the Moon always be 'there'? I'm not sure of that.

Period? Off hand I don't know, but if you take the time from Earth orbit to the Moon with Apollo, double it and add a bit you shouldn't be far off.


XF,
Actually that is not quite correct. If you read the Schwaniger etc link and explanation I cited below (https://en.wikipedia.org/wiki/Free-return_trajectory) you will see that they used that type of free-return orbit for Apollo 8, 10, 11, but then switched to a different (hybrid) orbit for Apollo 12 through until 17.
Regards,
dspp

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Re: Earth-Moon orbit

#292146

Postby 9873210 » March 18th, 2020, 11:22 pm

GoSeigen wrote:And the next question was about how useful such a satellite might be. I'm guessing not very useful, because you might as well just put a satellite in moon orbit. But what if it were something like the ISS: would there be any advantage to passing by the moon on a regular basis?

One possible use would be to return huge amounts of data from the moon. Radio or optical links of a few hundred kilometers allow far higher data rates with smaller amplifiers and antennae than a link of a few hundred thousand kilometers.

TESS does something of this sort, although far less extreme.

Never underestimate the bandwidth of a satellite full of solid state drives hurtling through space.

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Re: Earth-Moon orbit

#292168

Postby GoSeigen » March 19th, 2020, 6:37 am

9873210 wrote:
One possible use would be to return huge amounts of data from the moon. Radio or optical links of a few hundred kilometers allow far higher data rates with smaller amplifiers and antennae than a link of a few hundred thousand kilometers.


Hmm yes, that's a very good use case. Google Moon and Crater View. I like it.

Thank you to everyone for their thoughts about the orbit too. Brought back a lot of my A level physics, the most obvious point that I'd forgotten being that gravity is a conservative field so potential energy change is zero when you leave a point and then return to the same place, no matter which path is taken.

Thanks to dspp for the interesting and informative links.


GS

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Re: Earth-Moon orbit

#292802

Postby gryffron » March 20th, 2020, 11:44 pm

There are Lagrange points in the earth/moon orbits, which are eternally stable orbital points. But they don't go around the moon.

It's generally considered to be not a great idea to site objects at Lagrange points because they attract space debris.

Gryff

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Re: Earth-Moon orbit

#295140

Postby scotia » March 28th, 2020, 3:53 pm

dspp wrote:
What you are seeking is a special case of a special case, i.e. Trans-Lunar Injection orbit with a free return trajectory, that then repeats with no further orbital injection burns (maybe just trim). So the orbit needs to have a period of approximately 28 days or a multiple thereof, and will of course be elliptical to fit these constraints. It might be either a simple egg shaped, or figure-8 shaped, but I think the geometricians would consider them both to be capable of being calculated as ellipses (I'm not sure). However this indeed is possible, with various monthly periods, and I will let wiki give you the explanation of the ones Schwaniger calculated (see my last link below for diagrams etc).
regards, dspp

About 35 years ago, I came across the differential equations for such a case. They assumed Earth & Moon in circular orbit, that the gravitational force of the satellite on the Earth and Moon is negligible, and that all of the orbits are in the same plane. A rotating coo-ordinate system is used with the x-axis being the line joining the Earth to the Moon, and the centre of mass being the origin. The net result is a 4th Order differential equation system. The gravitational force follows the inverse square law, so these 4 first order differential equations are extremely non-linear.
I was teaching a course on the mathematical modelling of dynamic systems, and was preparing experiments to illustrate different numerical integration techniques - and the extremely non-linear nature of this problem highlighted the advantage of a variable step integration procedure - with comparatively short steps being essential when the satellite was near to the Earth or Moon, whereas they could be lengthened, so reducing computing time, when distant from either. An advantage of this satellite orbiting system is that the energy is conserved, so checking the energy at the beginning and end of the simulations will show up any numerical inaccuracies. I had developed a package in Turbo Pascal which allowed the students to solve these equations, and display the orbits on-screen and/or send them to a plotter. So a few days ago I had a look back at this work, and resurrected part of it in VB Net. Yes - you can select initial states that result in stable, repeating orbits with the same period as the Earth - Moon orbit. There are some interesting Orbit shapes! I'll capture a selection and produce some jpg files - to which I'll post a link. I used a variable step length Runge-Kutta integration method based on the example in the well known Numerical Recipes book, but I can't remember where I got the Earth-Moon orbiter equations from.

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Re: Earth-Moon orbit

#295142

Postby Itsallaguess » March 28th, 2020, 3:58 pm

Slightly off-topic, but related nonetheless so I hope it can be indulged...

I still struggle to believe that if we took all the other major planets in our Solar System and lined them up next to each other, they'd all fit within the orbital space between the Earth and the Moon -

Image

Source - https://futurism.com/you-can-fit-all-of ... d-the-moon

Cheers,

Itsallaguess

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Re: Earth-Moon orbit

#295143

Postby kiloran » March 28th, 2020, 4:04 pm

Itsallaguess wrote:Slightly off-topic, but related nonetheless so I hope it can be indulged...

I still struggle to believe that if we took all the other major planets in our Solar System and lined them up next to each other, they'd all fit within the orbital space between the Earth and the Moon -

Image

Source - https://futurism.com/you-can-fit-all-of ... d-the-moon

Cheers,

Itsallaguess

But wouldn't that somewhat muck up the dynamics of the solar system? As well as scotia's differential equations!
We would get some interesting solar eclipses, mind.

--kiloran

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Re: Earth-Moon orbit

#295184

Postby XFool » March 28th, 2020, 6:36 pm

Itsallaguess wrote:Slightly off-topic, but related nonetheless so I hope it can be indulged...

I still struggle to believe that if we took all the other major planets in our Solar System and lined them up next to each other, they'd all fit within the orbital space between the Earth and the Moon

Great! I've never seen that before, though I know and have seen it demonstrated that most people automatically greatly underestimate the distance between the Earth and the Moon on a scale basis.

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Re: Earth-Moon orbit

#295241

Postby scotia » March 28th, 2020, 10:40 pm

As promised - here are the plots of the satellite orbiting round the moon and the earth. The Earth (larger red circle) and the Moon (smaller red circle) appear stationary - since we are operating in a rotating frame of reference which is synchronous with the moon's orbit. In each of the three examples, the satellite starts off directly behind the moon, at a distance from the moon of 20% of the earth-moon distance. The velocity in the x direction (directly towards Earth) is zero. The velocity in the y direction can be user selected. The first stable orbit shown in the first picture used the velocity provided by the author of the equations (who's name I have forgotten!).

Image

The second stable obit was discovered by trial and error - the starting velocity is approximately 50% higher than in the first example

Image

Both of these simulations were run for two complete moon-earth orbits (i.e. the satellite traversed these curves twice) - showing that these satellite orbits are repeatable with a period equal to the moon-earth orbit time.

On trying out velocity values between these two a variety of non-repeating orbits are observed. One such value scores a direct hit on the earth after about 23% of the moon's orbit period:

Image

At this point we are nearing gravitational infinity points - and unsurprisingly we get numerical instability if we proceed any further.

The educational point behind this simulation was to introduce students to the importance of a variable step length integration procedure. I don't know if this message got through - but they enjoyed themselves looking at all sorts of orbits.

dspp
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Re: Earth-Moon orbit

#295300

Postby dspp » March 29th, 2020, 10:38 am

scotia wrote:As promised - here are the plots of the satellite orbiting round the moon and the earth. ......

The second stable obit was discovered by trial and error - .....

The educational point behind this simulation was to introduce students to the importance of a variable step length integration procedure. I don't know if this message got through - but they enjoyed themselves looking at all sorts of orbits.


Thanks, interesting. Takes me back to Stroud, Engineering Mathematics.

You could extend this problem set to include an optimisation algorithm. It would have to seek quite widely to find the different orbits, and the decision about which optimisation approach to go for might be interesting. A further refinement would be to have a nested optimiser running the Runga Kutta timestep decisions.

regards, dspp


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