Thursday, May 28, 2015

Orbital Momentum as Commodity

To The Moon, Alice

A tether needs to be substantially more massive than it's payloads. Else a catch/throw would wreck the tether orbit.

In my last post I described how near earth asteroids parked in retrograde lunar orbits might be used as anchors and momentum banks for lunar orbital tethers.

What could we use to anchor a vertical tether in earth orbit?

When a satellite in geosynchronous orbit satellite dies, it's sent to a graveyard orbit about 300 km above geosynch. According to IO9 there are more than a hundred sats in this graveyard. There will be more as time goes on.

Boost one of these dead sats 25 km higher than the graveyard orbit and dangle a 25 km tether. The tether foot will be moving about 10 kilometers/hour or about 5.5 mph with regard to the sat graveyard. As satellites are caught, momentum can be added via ion engines. With an ion engine's high ISP, the satellite collector's reaction mass can be small with regard to the mass of the satellites collected.

There's little plane change delta V since most the GEO sats are equatorial.

This would cleanse the geosynch neighborhood of orbital debris. When the dead sats are collected into a single mass, cross section is much smaller than a multitude of satellites. Thus likelihood of debris generating impacts is much smaller.

Some parts of these dead sats may be salvageable. Many may still have working solar arrays, for example. Although it is possible salvage issues could be a road block to this scheme. Those parts not salvageable would still have value just as a source of orbital momentum.

Boost this ball of dead sats to an orbit 10,000 km higher than GEO and extend a tether 10,000 km up and 7,000 km down. Now we have a well anchored 17,000 km tether. Tether foot is 3,000 km above geosynch so there's no chance the foot will hit an GEO sat. There is also less chance of tether damage from an impact with orbital debris.

A payload released from this tether top will reach the moon.

If we use Kevlar with a tensile strength of 3.6 giga pascals and density of 1.44 grams per cubic centimeter, taper ratio for this tether is 1.18. Tether to payload mass ratio is less than 1.

The tether center is moving about 2.78 km/s. Grave yard sats move about 3.06 km/s. So delta V to raise the ball of sats is only .28 km/s.

From LEO it takes around 3.2 km/s to rendezvous with this tether foot. About what it takes for a normal Trans Lunar Insertion from LEO. This was disappointing to me.

However this tether could receive mass from an asteroid parked in a lunar orbit. It could then send asteroidal mass to lower orbits.

If lower tethers were receiving mass from above as well as from earth, it might be less costly to give the lower tethers a substantial anchor mass.

Here are three possible tethers: the super GEO tether already described, a sub MEO tether and a super LEO tether:

Each of these tethers is positioned to avoid the high satellite/debris density areas: GEO (Geosynchronous Earth Orbit), MEO (Middle Earth Orbit, home of GPS and other global positioning sats), and LEO (Low Earth Orbit).

Tether to payload mass ratio is also less than one for both the Super LEO and sub MEO tethers.

The apogees and perigees of these red ellipses match the tether velocities at rendezvous. So almost no propellent would be used for catches and throws.

From LEO it takes .5 km/s to reach the Super LEO Tether. From there the relay of tethers can deliver spacecraft to the moon.

Once In The Moon's Neighborhood...

The Super GEO tether tosses payloads to a 384,400 km apogee. 384,400 km is the moon's distance from earth. Velocity of this ellipse's apogee is about .54 km/s while the moon's speed is about 1.01 km/s. Vinf with regard to the moon is about .48 km/s.

In my earlier post I described an asteroid anchored tether that could toss stuff from the moon with a Vinfinity of 1.6 km/s. Tether to payload mass ratio would be around 3. If we only need a Vinfinity of .48 km/s, a less massive tether would do:

Tether center's 24,200 km from the moon's center. Tether foot 13740 km, tether top 39000 km. If we use Kevlar with 3.6 giga pascal tensile strength and 1.44 g/cm^3 density, the tether to payload mass ratio is less than 1/4.

Dropping from this tether foot, a payload would impact the moon at 2.23 km/s.

It'd also be somewhat easier to park a near earth asteroid in this higher lunar orbit. See the Keck Report for a proposed method to park rocks in lunar orbit. For asteroids with orbital energy just above zero it is doable to park them in an high lunar orbit.

Up Momentum as a Valuable Commodity

Catching from a lower orbit and throwing upwards will sap a tether's orbital momentum. Earlier I had mentioned that ion engines using xenon as reaction mass could restore momentum.

But asteroidal mass from above is a source of up momentum.

With two way traffic, the need for xenon is reduced. Momentum boosting maneuvers could be balanced with momentum sapping catches and throws. The need for reaction mass would be largely eliminated.

An asteroid anchored tether in lunar orbit would be helpful in capturing more asteroids to lunar orbit. There's a lot of near earth asteroids in accessible orbits, so there's a massive source of up momentum.

Electrodynamic Tethers

How about using Lorentz force to change orbital momentum? As an electron moves up (away from earth's center) it passes through earth's magnetic field and the tether is pushed east, boosting momentum. Sending electrons down will give a westward push, reducing momentum.

But this relies on one way current. If the circuit is closed, electrons move up as well as down and there's no net Lorentz force.

The ionosphere can close a circuit in low earth orbit. The tether can pick up electrons from the ionosphere as well as discharge electrons into the ionosphere.

But the tethers described here are above the ionosphere. Using Lorentz force doesn't seem to be an option.

But I don't know that much about electrodynamic tethers, I could be wrong. If someone corrected me, it'd be a pleasant surprise.

Why I Don't Like Rotovators

Why my obsession with vertical tethers? Yes, rotovators could be shorter. But there's several reasons I don't like rotovators.

Taper Ratio and Tether to Payload Mass Ratio

Tension in rotovators comes from so called centrifugal force, ω2r. At the Space Stack Exchange 2012rcampion takes a look at a sling's tether taper ratios. Campion's results look a lot like Moravec's equations.

As tether tip speed grows, taper ratio soars. Short tethers capable of a good throw would be quite massive. According to this pdf, a LEO to GTO rotovator capable of tossing 5 tonne  payloads would need to mass 50 tonnes (3rd paragraph, page 4). A 10 to 1 tether to payload mass ratio.

In contrast, a vertical tether's tension comes from centrifugal force and gravity, ω2r - μ/r2. Gravity mitigates stress from centrifugal force and taper ratio grows a lot more slowly. All the vertical tethers described in this post have a tether to payload mass ratio less than 1.

Vertical tethers do need an anchor mass. But anchor mass can be very useful. I'd like to see lots of solar arrays at the tether centers. Solar arrays could power electrolysis plants to crack water into bi-propellent, move elevator cars up and down, and occasionally power Hall Thrusters to adjust the tether's orbit.


A rotovator is good for catches and throws when it's vertically aligned. But it's well aligned only for a very brief time during its spin. The rotovator must be correctly positioned when a launch window occurs. Ditto for catching from a Hohmann orbit.

Catches are harder. As a payload approaches a rotovator, the tip remains on its path for a brief time and then either zooms up or down (from the payload's point of view). In contrast, the tips of a vertical elevator remain at constant altitude.

Attitude Adjustment

A vertical tether stays vertical due to tidal acceleration gradient. It seems to me a rotovators attitude would need to be adjusted from time to time.

Possible Imports to Earth and LEO


I believe the most important import from asteroids will be water.

My last two blog posts are more or less in response to Jon Goff's The Slings and Arrows of Outrageous Lunar Transportation Schemes: Part 1 - Gear Ratios. Goff pointed out that only a small fraction of propellent mined at the lunar poles could be delivered to LEO.

Ever since Goff wrote that, I've been trying to think of ways to deliver extraterrestrial propellent to where it's needed.

"Wait a minute," you might be thinking, "This guy has just described a transportation system using momentum exchange and ion engines. Why is he still stuck on stone age chemical propellent?"

I believe the biggest obstacle to fully reusable spacecraft is an upper stage's 8 km/s re-entry. Hall Thrusters decelerate too slowly to help with that plunge.

A tether foot low enough to drop payloads into low velocity suborbital paths would be vulnerable to collision. Below 1000 km, space is full of sats and debris.

What the Super LEO tether could do is deliver propellent to LEO. An ellipse from the Super LEO foot would take .5 km/s to circularize at perigee. And some of that .5 km/s might be accomplished by aerobraking.

An upper stage refueled at LEO could do a healthy burn to lose most the 8 km/s. The upper stage might also beef up it's dry mass with structural support and TPS, also from asteroids. Given these options, upper stage re-use is very doable. If upper as well as booster stages can be economically re-used, the dream of cheap space access is realized.


Many asteroids have high concentrations of the platinum group metals. Right now these are precious due to rarity. But should they become more available, there are numerous ways they can be used.

Rare earth metals have many uses. Rare earths actually aren't rare. But they're hard to mine in an eco-friendly way. I would much rather see them mined on a lifeless, barren rock than in earth's biosphere. An asteroid in lunar orbit would make the moon's KREEP more accessible. Besides rare earths, KREEP also contains uranium and thorium, possible sources of energy.


As mentioned earlier, imported uranium and thorium could fuel terrestrial power plants.

The earth intercepts only .45 billionths of the sun's light. We're using only a tiny fraction of possible solar energy.

If we use solar energy to refine extra-terrestrial ore and import commodities to earth and LEO, in a sense we're importing energy.

Besides energy for refining, imported commodities would also require energy for transportation. To get a kilogram from just above C3 to LEO takes about 512 mega-watts. Tether momentum exchange could accomplish most of this. But it's energy used, regardless of source.


A source of up momentum would be a major game changer. A first step towards acquiring this source would have been the early version of the Asteroid Redirect Mission (ARM). This is doable. But for now it looks like popular opinion will keep this from ever being funded.

I'll continue singing the praises of asteroids and ARM. God willing, my small voice will have some influence.

Wednesday, May 20, 2015

Have your ion ISP and eat Oberth cake too.

Early versions of the Asteroid Redirect Mission suggested retrieving an asteroid in heliocentric orbit and parking it in a lunar Distant Retrograde Orbit (DRO).

Retrograde lunar orbits with apolunes below 40,000 km can be stable for centuries without station keeping. Once we park an asteroid in such an orbit, it's not going anywhere. Once we get higher than about 3/5 of a Hill Sphere radius, the orbits are more fragile.

The proposed asteroid retrieval vehicle would be powered by solar arrays and propelled by Hall Thrusters using xenon as reaction mass.

A 500 tonne rock in retrograde lunar orbit along with solar panels and Hall Thrusters would make a great anchor for a tether. It would also serve as a momentum bank.

Hall Thrusters could build the tether's momentum gradually. But catching or releasing a payload is a sudden event equivalent to an impulsive burn from a high thrust chemical rocket.

Here is one possible vertical tether whose foot is 300 km above the lunar surface:

This ~6000 km tether is a lot shorter than a full blown Clarke tower. The moon's gravity well is also shallower than earth's. So this tether endures much less stress. Kevlar would suffice for the tether material and taper ratio would be about 1.4

This tether top is moving about 1.95 km/s. Releasing a payload from this tip would send it into a hyperbolic orbit with regard to the moon with a Vinfinity of about 1.6 km/s. This gives a multitude of possible orbits depending on what barycentric longitude the payload is released at.

Here is the path when released at 129º barycentric longitude, no burns:

Flight time from release to perigee is about 51 hours. At perigee the speed is ~10.9 km/s, just above earth escape. This is a hyperbolic path with regard to earth with a Vinfinity of about .3 km/s. Doing a burn at this perigee would confer a large Oberth benefit. A .4 km/s burn would suffice for Trans Mars Insertion.

Here is release from the same tether but at 0º:

In this case the Vinfinity vector is nearly parallel to the moon's velocity vector. So the 1.6 km/s Vinf is added to the moon's velocity vector for a total of about 2.6 km/s with regard to the earth. Vinf with regard to the earth is about 2.17 km/s. This is more than enough for insertion to transfer orbits to most near earth asteroids.

A .62 km/s burn just after release would boost lunar Vinf to 2.33 km. If released so Vinf is parallel to the moon's velocity vector, speed wrt earth would be 3.33 km/s. This gives a 3 km/s Vinf, enough to reach a 1.52 aphelion. (1.52 A.U. is semi major axis of Mars' orbit).

Something dropped from the foot of this tether would impact the moon's surface at 1.04 km/s. A circular low lunar orbit is about 1.6 km/s. So it would would make lunar landing somewhat easier. Sadly, this would only be useful for lunar sites on lower latitudes.

From earth you can lob a payload into a retrograde hyperbola about the moon. Just set apogee above the moon. This transfer orbit takes a LEO burn of about 3.1 km/s. From LEO to apogee is about 5.5 days.

For this transfer orbit, Lunar Vinf is about .83 km/s.

There is a point on the tether that release payload into a hyperbola with this Vinf:

If the transfer orbit is coplanar with the tether, rendezvous with the TransEarth Bead would cost zero delta V if we had perfect control. Practically speaking, I'd expect some tweaking of the orbit, especially during the final approach.

Releasing from the TransEarth Bead can also send a spacecraft back to earth along an elliptical orbit. Tether departure zero delta V. Circularizing at LEO would take 3.1 km/s. Some or most of the circularization delta V could be accomplished with aerobraking.

So a lunar tether with an asteroid anchor can be just 3.1 km/s from earth. Some carbonaceous asteroids are 20% water by mass. Also carbon compounds. They could potentially supply life support consumables, radiation shielding and propellent. If much of the spacecraft's mass comes from the asteroid, that greatly reduces lift off mass from earth's surface.

There are many asteroids in earth like orbits whose C3 with regard to earth is just above zero. These can be parked in a lunar retrograde orbit using modest sized spacecraft. See the Keck report. A ordinary rocket like the Atlas V could launch retrieval vehicles capable of bringing back 500 tonne rocks.

At the moment one of my favorite daydreams is a partnership between Planetary Resources and Tethers Unlimited. I believe it's doable to open up the solar system.

Tuesday, May 19, 2015


Earth Moon Lagrange 2 or EML2 is one of 5 locations where earth's gravity, moon's gravity and so called centrifugal force all cancel out. It lies beyond the far side of the moon at about 7/6 of a lunar distance from earth.

Infrastructure at any of these 5 locations could be kept in place with a small station keeping expense. Other high earth orbits would be destabilized by the influences of the earth, moon or sun.

Of these 5 locations, EML2 is the closest to escape. How close?

Specific orbital energy is given by

v2/2 - GM/r

v: velocity with regards to the earth
G: gravitational constant
M: mass earth
r: distance from earth center

Here's specific orbital energies for a few orbits:

For EML1 and EML2 I'm looking at resulting earth orbits for payloads nudged away from Luna's Hill Sphere.

Most the energy is getting from earth's surface to Low Earth Orbit (LEO). Then another huge chunk is getting from LEO to escape.

EML2 is right next door to escape (aka C3=0). If the goal line is Trans Mars Injection, EML2 is on the 9 yard line.

EML2's orbital energy is about -180,000 joules per kilogram. How much is that? Well, Kattie is standing next to the small generator which provides electricity for our business during power outages during the summer monsoons. It would take this 20 kilo-watt generator 9 seconds to crank out 180,000 joules.

An EML2 payload nudged away from Luna would rise to an 1.8 million km apogee. An ordinary earth orbit at 450,000 kilometers from earth's center would move about .94 km/s. But since EML2 is moving at the moon's angular velocity, it is traveling 1.19 km/s.  Earth's Hill Sphere is about 1.5 million km in radius. So depending on timing, an EML2 nudge could send a payload out of earth's sphere of influence into a heliocentric orbit.

Another possibility is the sun's influence could send a payload back towards the earth with a lower perigee:

All of these pellets were nudged from EML2. The sun's influence has wrested most of these from earth's influence. But check out pellet number 3 (orange). The sun's influence has dropped this pellet to a perigee deep in earth's gravity well. For a .1 km/s nudge from EML2 we can get a deep perigee that can give a very healthy Oberth benefit. However, such a route takes  about 100 days.

Using an lunar gravity assist along with an Oberth enhanced burn deep in the moon's gravity well, EML2 is 9 days and 3.5 km/s from Low Earth Orbit (LEO):

This route was found by Robert Farquhar.

Bi-Elliptic Transfers

It radii of two different orbits differ by a factor of 11.94 or more, a bi-elliptic transfer takes less delta V than Hohmann. EML2 radius / LEO radius is about 67, so LEO to EML2 could definitely be a beneficiary of bi-elliptic.

From LEO, a 3.1 km/s burn gets us to a hair under a escape. A multitude of elliptical orbits fall under this umbrella!

As you can see, it takes almost as much to get as high as EML1 as it does to reach a 1.8 million km apogee. I chose 1.8 million as an apogee since a 450,000 x 1,800,000 km ellipse at perigee has the same altitude and speed as EML2. At perigee a payload can slide right into EML2 with little or no parking burn.

What's needed is an apogee burn to raise perigee to 450,000 km. A 6738x1,800,000 km ellipses moves very slow at apogee, a mere .04 km/s. A 450,000 x 1,800,000 km ellipse doesn't move much faster at apogee, about .3 km/s So a .26 km/s apogee burn suffices to raise perigee.

So the total budget is .26 + 3.1555 km/s. This 3.42 km/s delta V budget is better than a Hohmann but about the same as Farquhar's 9 day route.

But recall apogee is beyond earth's Hill Sphere. With good timing, the sun can provide the apogee delta v.

Here's a route I found with my shotgun orbital sim:

LEO burn is about 3.11 km/s. Payload passes near the moon on the way out, boosting apogee and rotating line of apsides. The sun boosts apogee as well as perigee. Coming back the pellets slide right into EML2 (the circular path alongside the Moon's orbit).

This LEO to EML2 route took 74 days and 3.11 km/s.

EML2 and Reusable Earth Departure Stages.

Using the Farquhar route, it takes about .4 km/s to drop from EML2 to a perigee moving just under escape velocity. At this perigee .5 km/s will give Trans Mars Insertion (TMI). After the departure stage separates from the payload it's pushing, it can do a .5 km/s braking burn to drop to an ellipse with a near moon apogee. Once at the moon, another .4 km/s takes the EDS back to EML2.

For massive craft moving from between earth's neighborhood and other heliocentric orbits, it makes little sense to climb down to Low Earth Orbit (LEO) and back each trip. It saves time and and delta V to park at EML2 on arrival. If EML2 becomes a stop for interplanetary space craft, a reusable EDS is a good way to depart the earth/moon neighborhood.

I talk about this in more detail at Reusable Earth Departure Stages.

EML2 and Fast Transits

Here's a pic of a Non Hohmann Mars transfer:

Mars and earth orbits are approximated as circular orbits. A Hohmann orbit will have a 1 A.U. perihelion and a 1.52 A.U. aphelion. The transfer orbit above has perihelion .7 A.U. and aphelion 1.53 A.U. Semi-major axis of this orbit is (.7 + 1.53)/2  A.U. or 1.115 A.U. Orbital period is 1.1153/2 years which is about 1.18 years.

The trip to Mars isn't the entire orbital period though, just the turquoise area swept out from departure to destination. The turquoise area is 31.5% of the ellipse's area. 31.5% of 1.18 years is about 135 days or about 4.4 months.

Departure and arrival Vinf are indicated by the red arrows. These are the vector differences between the transfer orbit's velocity vector and the planet's velocity vector at flyby. A change in direction accounts for most of the Vinf. I'm assuming Mars and Earth are in circular orbits with a zero flight path angle. Therefore the direction difference between vectors can be described with the flight path angle of the transfer orbit's velocity vector.

In this case the earth departure Vinf is 11.3 km/s. That's a big Vinf! But if falling from EML2, only a 4.9 km/s perigee burn is needed. This is doable.

At Mars the Vinf is 5.14 km/s. But a periaerion burn of 2.57 km/s brakes the orbit into an (3697x2345 km ellipse. This orbit could be circularized via periaerion drag passes through the upper atmosphere. Since this ellipse has a period less than a day, orbit could be circularized in a few weeks.

An upper stage can have a 8 km/s delta V budget. Recall it takes about .4 km/s to fall from EML2. Therefore let's try to find a route that takes about 7.6 km/s from perigee burn to periaerion burn.

Trial and error with my Non Hohmann Transfers spreadsheet gives:

With chemical rockets departing from EML2, I believe 4 month trips to Mars are doable.

EML2 Proximity to Possible Propellent Sources.

In terms of delta V, time and distance EML2 is quite close to several possible propellent sources.

There are thought to be frozen volatiles in the lunar cold traps. Some craters at the lunar poles have floors in permanent shadow. Temperatures can go as low as 30 K. Volatiles that find their way to the cold traps would freeze out and remain. There may be rich deposits of H20, CO2, CH4, NH3 and other compounds of hydrogen, carbon, oxygen and nitrogen. These would be valuable for life support as well as propellent.

The moon's surface is about 2.5 km/s from EML2.

Also there are proposals to retrieve asteroids and park them in lunar Deep Retrograde Orbits (DROs). DROs are stable lunar orbits that can remain for centuries without station keeping. Planetary Resources would like to retrieve water rich carbonaceous asteroids. Carbonaceous asteroids can contain up to 20% water by mass in the form of hydrated clays. They can also contain compounds of carbon and oxygen.

LDROs would be about .4 km/s from EML2.


EML2 would make a great transportation hub. Not only for travel to destinations throughout the solar system but also within our own earth moon neighborhood.

Wednesday, May 6, 2015

The Need for a Better Alpha

What the heck is Alpha?

What Alpha am I talking about? A power source's ratio of mass to power. The first time I ran into this quantity was in a NASASpaceFlight discussion of VASIMR's 39 day trip to Mars.

Kirk Sorensen worked for NASA 10 years. He has Master's degrees in aerospace as well as nuclear engineering. He is co-founder of Flibe Energy, a company that hopes to build thorium nuclear reactors. I don't always agree with him but I believe he has some expertise in this field. For the time being I am talking his word for it.

So this Magic Alpha, .50 kg/kWe, what is that? Here's an attempt to portray it:

This Ford Focus has a 160 horsepower engine. One horsepower = ~750 watts, so the engine is about a 120,000 watt power source. Not only the engine but also gasoline and oxygen. Pictured with the Focus is Dominique who masses about 60 kilograms. If she were a power source replacing the engine, gasoline and oxygen, we'd have an alpha of .5 kg/kW.

An electric car like the Tesla uses a battery. But just as a gas engine must make periodic stops to gas up, a Tesla must be frequently recharged. Enroute to Mars there are no gas stations and no electric outlets.

Later in the same NASA Space Flight thread, Sorensen says:
I don't really care if Samim Anghaie is crook but his 0.5kg/kWe number is a fantasy. Why FCD builds his VASIMR sales case on that number when all other reputable electric propulsion researchers have rejected it (even though makes their thrusters look incredible too) is beyond my understanding.

Indeed. Such a power source would make Hall thrusters look great. So far as I know Franklin Chang Diaz and Samim Anghaie are the only folks whose schemes rely on such an alpha.

Thermal Watts vs Electric Watts

39 day VASIMR trips to Mars are mentioned on Page 42 of The Plundering of NASA by Rick Boozer. Boozer argues that SLS and Orion are pork barrel make work programs and that money could be better spent on SpaceX and other programs. In general I agree with Boozer but was disappointed to see his endorsement of VASIMR.

So I asked Boozer about the Magic Alpha. Boozer came back with Project Nerva, a nuclear thermal rocket. He wrote:
Project NERVA claimed up to 5 GW possible with total mass of 38,600 kg. That works out to .00008 kg per W or .008 per kW. According to that Sorenson is incorrect.

NERVA's output is thermal watts. Thermal and electric watts are two very different things.

A nuclear electric power plant must first convert thermal watts to electric watts. But that's not the only problem.

The plant must dump waste heat. Massive cooling towers have become an icon for nuclear energy. From  Wikimedia:

Earthly nuclear power plants can use water to carry off waste heat. In space there are no neighboring streams a nuclear power plant can use. In fact vacuum is a great insulator. A nuclear electric power source would need massive radiators.

I mentioned to Boozer that thermal and electric power sources were very different things.

He replied:
David, I just don't know where you are coming from. There you went earlier lecturing me about the difference between thermal and electrical energy which is something that I teach physics students all the time. If I wasn't competent in physics my students wouldn't be making the high grades they are and I couldn't have got my Master's in astrophysics.
I was hoping Boozer would demonstrate Sorensen was wrong. Sadly, pointing to his students' good grades and his degree did absolutely nothing to demonstrate the plausibility of Diaz' Magic Alpha. I was convinced of one thing though: Rick Boozer isn't credible.

I did not bother reading past page 42 of The Plundering of NASA.

What's the best plausible Alpha?

I search space forums for discussions of low mass power sources. So far as I can tell, thin film photovoltaics show the best promise. Roll Out and Passively Deployed Array (RAPDAR) might deliver 250 kW/kg. RAPDAR's thin film solar cells use an Elastic Memory Composite (EMC) for support and structure. Rolled up and cooled, the EMC will fit in a small volume and thus can fit under a fairing. When the sun warms it, the EMC will expand to the shape it needs to be.

On a Nasa Space Flight thread space entrepreneur Jeff Greason opined:

While I won't speak to this specific design, more generally I am quite convinced that thin film solar approaches 1 kW/kg are definitely possible near term. However there is very little serious work going on, and packaging such systems for launch and deploying them without spoiling the mass is not at all trivial. 
But do keep thinking -- it is not crazy, at least 1 kW/kg rather than two. 
Thin film solar is extremely fragile, however, so the packaging is really challenging.
That's the rub, packaging. How useful are acres of Saran Wrap® with no structure? There needs to be a supporting frame to keep the film spread. It also needs to be kept pointing towards the sun so the supporting frame needs to be attached to gimbals and motors. What is the Alpha including supporting structure, gimbals and motors?

How will we deploy acres of Saran Wrap® from a small volume that fits within a fairing?

However Greason's optimism is somewhat reassuring. Being a bonafide space-cadet, I cling to optimistic opinions as long as I can.

Why is Alpha such a big deal?

As mentioned at the beginning of this post, a great alpha would make ion thrusters a more formidable tool. Presently ion thrusters have great ISP but very slow acceleration. A big cut to parasitic mass would give ion thrusters better acceleration.

Good Alpha would also make ISRU more plausible. Readers of my blog know I'm hung ho on use of extra-terrestrial propellent, either near earth carbonaceous asteroids or frozen volatiles in the lunar cold traps.

Let's say we do mine water in the moon's neighborhood and we want to crack it to hydrogen/oxygen bi-propellent. Cracking a mole of water (18 grams) takes 287000 joules. A tonne of water is 55555 moles. 55555 moles*287000 joules/mole =13166666667 joules. If we wanted to crack 10 tonnes of water per day, we'd need a 1.5 mega-watt power source. And that doesn't include refrigerating the cryogens.

Cracking water isn't the only ISRU electricity hog. Just about all extra-terrestrial mining and industry will need lots of juice.

In the 50's and 60's NASA and the military provided big incentives to miniaturize electronics as low mass and small volume circuitry is a pre-requisite for rockets and missiles. I believe miniaturizing a power source should be a top goal for NASA. If we hope to settle space, a better Alpha should be given a higher priority than Apollo redux.

Sunday, May 3, 2015

A Golden Escher tribute

Escher loved spirals, the Droste effect, and recursive stuff that suggests infinity.

I was looking at a golden rectangle with a series of squares marked off and thought it'd make for a neat variation on Escher's print of hands drawing hands.

Sunday, April 5, 2015

Potholes on the Interplanetary Superhighway.

Wikipedia describes the Interplanetary Transport Network as "… pathways through the Solar System that require very little energy for an object to follow." See this Wikipedia article. They also say "While they use very little energy, the transport can take a very long time."

Low energy paths that take a very long time? I often hear this parroted in space exploration forums and it always leaves me scratching my head.

The lowest energy path I know of to bodies in the inner solar system is the Hohmann orbit. Or if the destination is noticeably elliptical, a transfer orbit that is tangent to both the departure and destination orbit. Although I think of bitangential transfer orbits as a more general version of the Hohmann orbit.

Bitangential Transfer Orbit
The transfer orbit is tangent to both departure and destination orbit.
The Hohmann transfer is the special case where departure and destination orbits are circular.
Illustration from my pdf on tangent orbits.

In the case of Mars, a bitangential orbit is 8.5 months give or take a month or two. Is there a path that takes a lot longer and uses almost no energy? I know of no such path.

L1 and L2

The interplanetary Superhighway supposedly relies on weak stability or weak instability boundaries between L1 and/or L2 regions. Here is an online text on 3 body Mechanics and their use in space mission design. The authors are Koon, Lo, Marsden and Ross. Shane Ross is one of the more prominent evangelists spreading the gospel of the Interplanetary Super Highway.

The focus of this online textbook is the L1 and L2 regions. From page 10:

L1 and L2 are necks between realms. In the above illustration the central body is the sun, and orbiting body Jupiter. L1 and L2 are necks or gateways between three realms: the Sun realm, the Jupiter Realm and the exterior realm.

Travel between these realms can be accomplished by weak stability or weak instability boundaries that emanate from L1 or L2. From page 11 of the same textbook:

My terms for various Lagrange necks

First letter is the central body, the second letter is the orbiting body.

Earth Moon L1: EML1
Earth Moon L2: EML2

Sun Earth L1: SEL1
Sun Earth L2: SEL2

Sun Mars L1: SML1
Sun Mars L2: SML2

Since I'm a lazy typist that is what I'll use for the rest of this post.

EML1 and 2

I am very excited about the earth-moon Lagrange necks. They've been prominent in many of my blog posts and I plan to devote another blog post to EML2 soon.

EML1 and 2 are about 5/6 and 7/6 of a lunar distance from earth:

Both necks move at the same angular velocity as the moon. So EML1 moves substantially slower than an ordinary earth orbit would at that altitude. EML2 moves substantially faster.

It takes only a tiny nudge and send objects in these regions rolling about the slopes of the effective potential hills. Outside of the moon's influence they tend to fall into ordinary two body ellipses (for a short time).

Here's the ellipse an object moving at EML1 velocity and altitude would follow if the moon weren't there:

An object nudged earthward from EML will fall into what I call an olive orbit.
It's approximately 100,000 x 300,000 km.

In practice an EML1 object nudged earthward will near the moon on the fifth apogee. If coming from behind, the moon's gravitational tug can slow the object which lowers perigee.

Here is an orbital sim where the moon's influence lowered perigee four times:

I've run sims where repeated lunar tugs have lowered perigees to atmosphere grazing perigees. Once perigee passes through the upper atmosphere, we can use aerobraking to circularize the orbit.

Orbits are time reversible. Could we use the lunar gravity assists to get from LEO to higher orbits? Unfortunately, aerobraking isn't time reversible. The atmosphere can't increase orbital speed to achieve a higher apogee. And low earth orbit has a substantially different Jacobi constant than those orbits dwelling closer to the borders of a Hill Sphere.

So to get to the lunar realm, we're stuck with the 3.1 km/s LEO burn needed to raise apogee. But once apogee is raised, many doors open.

There are low energy paths that lead from EML1 to EML2. EML2 is an exciting location.

Without the moon's influence, an object at EML2's velocity and altitude
would fly to an 1,800,000 km apogee. This is outside of earth's Hill Sphere!

In the above illustration I have an apogee beyond SEL2. But by timing the release from EML2, we could aim for other regions of the Hill Sphere, including SEL1.

Here is a sim where slightly different nudges send payloads from EML2:

See how the sun bends the path as apogee nears the Hill Sphere? From EML2 there are a multitude of wildly different paths we can choose. In this illustration I like pellet #3 (orange). It has a very low perigee that is moving about 10.8 km/s. And it got to this perigee with just a tiny nudge from EML2. Pellet # 4 is on it's way to a retrograde earth orbit. Most of the other pellets are saying good bye to earth's Hill Sphere.

I am enthusiastic about using EML1 and EML2 as hubs for travel about the earth-moon neighborhood. But a little less excited about travel about the solar system.

We've left Earth's Hill Sphere. Now what?

Recall that EML1 and 2 are ~5/6 and 7/6 of a lunar distance from the earth. SEL1 and 2 are much less dramatic: 99% and 101% of an A.U. from the sun. Objects released from these locations don't vary much from earth's orbit:

Running orbital sims gets pretty much the same result pictured above.

Mars is even worse:

Are there weak instability boundary transfers leading from SEL2 to SML1? I don't think this particular highway exists.

To get a 1.52 A.U. aphelion, we need a departure Vinfinity of 3 km/s. To be sure EML2 can help us out in achieving this Vinfinity. In other words we could use lunar assists to depart on a Hohmann orbit. But a Hohmann orbit is different from the tube of weak instability boundaries we're led to imagine.

And once we arrive at a 1.52 aphelion. we have an arrival 2.7 km/s Vinfinity we need to get rid of.

Pass through SML1 at 2.7 km/s and you'll be waving Mars goodbye. The Lagrange necks work their mojo on near parabolic orbits. And an earth to Mars Hohmann is decidedly hyperbolic with regard to Mars.

What about Phobos and Deimos? The Martian moons are too small to lend a helpful gravity assist. We need to get rid of the 2.7 km/s Vinf and neither SML1 nor the moons are going to do it for us.

Mars ballistic capture by Belbruno & Toppotu

Edward Belbruno is another well known evangelist for the Interplanetary Superhighway (though he likes to call them ballistic captures). Belbruno cowrote this pdf on ballistic Mars capture.

Here is a screen capture from the pdf:

The path from Earth@Departure to Xc is pretty much a Hohmann transfer. In fact they assume the usual departure for Mars burn. Arrival is a little different. They do a 2 km/s heliocentric circularization burn at Xc (which is above Mars' perihelion). This particular path takes an extra year or so to reach Mars.

So they accomplish Mars capture with a 2 km/s arrival burn. At first glance this seems like a .7 km/s improvement over the 2.7 km/s arrival Vinf.

Or it seems like an advantage to those unaware of the Oberth benefit. If making the burn deep in Mars' gravity well, capture can be achieved for as little as .7 km/s.

Comparing capture burns it's 2 km/s vs .7 km/s. So what do we get for an extra year of travel time? 1.3 km/s flushed down the toilet!

What About Ion Engines?

"What about ion engines?" a Belbruno defender might object. "They don't have the thrust to enjoy an Oberth benefit. So Belbruno's .7 km/s benefit is legit if your space craft is low thrust."

Belbruno & friends are looking at a trip from a nearly zero earth C3 to a nearly zero Mars C3. In other words from the edge of one Hill Sphere to another.

So to compare apples to apples I'll look at a Hohmann from SEL2 to SML1. I want to point out I'm not using Lagrange necks as key holes down some mysterious tube. They're simply the closest parts of neighboring Hill Spheres.

"Wait a minute..." says Belbruno's defender, "We're talking Hall thrusters. So no Hohmann ellipse, but a spiral."

Low earth orbit moves about 4º per minute. So a low-thrust burn lasting days does indeed result in a spiral. But Earth's heliocentric orbit moves about a degree per day while Mars' heliocentric orbit moves about half a degree per day. At this more leisurely pace, a 4 or 5 day burn looks more an impulsive burn. The transfer between Hill Spheres is more Hohmann-like than the spiral out of earth's gravity well.

Instead of a 1 x 1.524 AU orbit, the new Hohmann is  a 1.01 x 1.517 AU ellipse. The new Hohmann's perihelion is a little slower, the new aphelion a little faster.

Moreover, SEL2 moves at the same angular velocity as earth. So it's speed is about 101% earth's speed. Likewise SML1 moves at about 99.6% Mars' speed.

With this revised scenario, aphelion rendezvous delta V is now more like 2.4 km/s. Still, Belbruno's 2 km/s capture burn saves .4 km/s.

.4 km/s is better than chopped liver, right? Well, recall ion engines with very good ISP. I'll look at an exhaust velocity of of 30 km/s.

e2.4/30 - 1 = .083
e2/30 - 1 = .069

So given a 100 tonne payload, rendezvous xenon is 8.3 tonnes for Hohmann vs 6.9 tonnes for Belbruno's ballistic capture.

108.3/106.9 = 1.0134

We're adding a year to our trip time for a one percent mass improvement? Sorry, I don't see this a great trade-off.


The virtually zero energy looooong trips between planets are an urban legend.

I'll be pleasantly surprised if I'm wrong. To convince me otherwise, show me the beef. Show me the zero energy trajectory from an earth Lagrange neck to a Mars Lagrange neck.

Until then I'll think of this post as a dose of Snopes for space cadets.

I'd like to thank Mike Loucks and John P. Corrico Jr. I've held these opinions for awhile but didn't have the confidence to voice them. Who am I but an amateur with no formal training? But talking with these guys I was pleasantly surprised to find some of my heretical views were shared by pros. Without their input I would not have had the guts to publish this post.

Friday, April 3, 2015

A spiral of tethers

First off let's look at the great granddaddy of vertical tethers, the Clarke tower.

For a vertical tether in circular orbit, there's a point where the net acceleration is zero. Above that point, so called centrifugal force exceeds gravity. Below that point, gravity exceeds so-called centrifugal force. If a payload is released on this point of on the tether, it will follow a circular orbit alongside the tether. This point I call the Tether Center.

In this case, the tether center is at geosynch height, about 42,000 km from earth's center. I set 42,000 km to be 1. What path does a payload follow if released from the tether below the center?

It will be a conic section. Call the conic's eccentricity e. Call the distance from tether point r.

If dropped from below center, r  = (1-e)1/3.
If released from above center, r  = (1+e)1/3.

Here's my derivation. Mark Adler also gives a nice demonstration in the comments on that post.

This is true of any vertical tether in a circular orbit.

If there are two prograde, coplanar vertical tethers at different altitudes, there's an elliptical path between them where the perigee velocity matches a point on the lower tether and apogee velocity matches a point on the upper tether.

If a payload is released from the lower tether at the correct time, it will rise to the upper tether which will be moving the same velocity as the payload at apoapsis. Rendezvous can be accomplished with almost no delta V. Cargo can be exchanged between tethers with almost no reaction mass.

Let r for the release point above the tether be (1+e)1/3 and release point below the tether be (1-e)1/3. Then both the larger and smaller ellipse will be the same shape.

Center of the above tether is 8000 km. I tried to place it above the dense orbital debris regions of low earth orbit. The tether is 461.6 kilometers long. Dropping from the foot will send a payload to a 150 km attitude perigee. Throwing a payload from the tether top will send a payload to a 9780 km apogee.

From a 150 km altitude orbit, it takes about .33 km/s to send a payload to the tether foot.

Both ellipses have the same eccentricity, about .0864

I repeatedly clone, scale by 126% and rotate 180º:

By ascending and playing catch with a series of tethers, a payload might make it's way from LEO to the vicinity of the moon:

But there's a problem with this scheme. A tether loses orbital momentum each time it catches a payload from below. Ascending and throwing to a higher orbit also saps orbital momentum. How do we keep these tethers from sinking?

Imagine resources parked in lunar orbit. Maybe propellent mined from the lunar poles. Or perhaps platinum from an asteroid parked in a lunar DRO. To send cargo to earth's surface or low earth orbit would entail catching from a higher orbit, descending and dropping to a lower orbit:

If cargo is moved down as well as up, momentum boosting maneuvers can be balanced with momentum sapping maneuvers.

Thus mass in high orbits are sources of up momentum. This itself could be a commodity, a way to preserve orbits of momentum exchange tethers.

This tether spiral scheme cuts tether length, especially in regions of high debris density and the Van Allen Belts.

In this illustration successive ellipses vary by a factor of 21/3. Other rates of expansion are possible. Let k be the ratio of one ellipse apogee to the apogee below. k = (1+e)4/3/(1-e)4/3. Thus we can wind the spiral tighter or loosen it by choice of ellipse eccentricity.