Wednesday, August 17, 2016

Tran Cislunar Railroad

Three Orbital Tethers

This post revisits Orbital Momentum As A Commodity. But now I will examine these tethers using Wolfe's spreadsheet.

I envision 3 equatorial tethers to move stuff back and forth between LEO and the lunar neighborhood:

The location of these vertical tethers avoids zones of orbital debris:

The orange regions, LEO, MEO and GEO, have high satellite and/or debris density. Thus tethers in those regions would be more vulnerable to damage from impacts.

Dead Sats for tether anchors

Unless elevator mass is lot more than the payloads, the acts of catching or throwing could destroy the tether orbit. At first it looks like the need for a substantial anchor mass is a show stopper. But there are a large number of dead sats in equatorial orbits. By one estimate,  there's 670 tonnes in the graveyard orbit above geosynch.

The dead sats gathered might have functioning solar arrays. According to this stack exchange discussion, solar arrays degrade by 2 to 3% a year due to radiation, debris impacts and thermal degradation. Thus a 20 year old array could still be providing 50% to 66% of the power it delivered at the beginning of its life. The parabolic dishes for high gain antennas might also be salvageable.

Whether functioning or not, solar arrays as well as other paneling might be used as shades to keep propellent cold. If our tethers receive propellent from the moon or from asteroids parked in lunar orbit, shades would help with cryogenic storage.

Consolidating dead equatorial satellites would reduce their cross sectional area and help solve the problem of orbital debris.

Super GEO tether

The circular orbit pictured above is 10,000 km above Geosynchronous Earth Orbit (GEO). The lower part of the tether has a length of 7,000 km and the upper tether is 10,340 km in length.

A Space Stack Exchange answer estimates there are 670 tonnes of dead sats in the geosynch graveyard orbit.

Delta V to raise the dead sats to this higher orbit is about .28 km/s. This might be accomplished with ion engines. Also the elevator could be used to send some to the sats towards the lower MEO tether. This would help with the .28 km/s delta V budget.

Upper Super GEO Tether, 10,340 km long
Safety Factor 3
Zylon taper ratio: 1.38
Tether to payload mass ratio: .78
Tether top radius 62,504 km
Tether top speed: 3.3 km/s
Tether top net acceleration: .07 m/s2 (.007 g)
Payload apogee: 384,400 km
Payload apogee speed: .53 km/s

The payload apogee is at lunar altitude and the payload's moving .53 km/s. The moon moves at about 1 km/s. So Vinf with regard to the moon is about .47.

Lower Super GEO tether, 7,100 km long
Safety Factor 3
Zylon taper ratio: 1.21
Tether to payload mass ratio: .47
Tether foot distance from earth 45,000 km
Tether foot speed: 2.4 km/s
Tether foot net acceleration: .07 m/s2 (.007 g)
Payload perigee: 21,450 km
Payload perigee speed: 5 km/s

The tether foot drops a payload to rendezvous with the MEO tether.

Sub MEO Tether

The circular orbit of the Sub MEO anchor mass is has a radius of 19,425 km. To get satellites from the super synchronous graveyard orbit to this orbit takes about 1.4 km/s. Some of that 1.4 km/s might be accomplished with the super GEO tether. Sending mass downward would help push the remaining GEO sats upward.

Upper Sub MEO Tether, 2,050 km long
Safety Factor 3
Zylon taper ratio: 1.30
Tether to payload mass ratio: .61
Tether top distance from earth 21,450 km
Tether top speed: 5 km/s
Tether top net acceleration: .3 m/s2 (.03 g)
Payload apogee: 45,000 km
Payload apogee speed: 2.4 km/s

The payload apogee radius and speed matches the foot of the super  GEO tether's radius and speed.
The top of this tether's radius and speed matches the payload perigee and speed sent from super GEO tether. The Sub MEO and Super GEO tethers can exchange payloads with minimal delta V at tether/payload rendezvous.

Lower Sub MEO tether.
Safety Factor 3
Zylon taper ratio: 1.35
Tether to payload mass ratio: .78
Tether foot radius 17,375 km
Tether foot speed: 4.1 km/s
Tether foot net acceleration: .38 m/s2 (.038 g)
Payload perigee: 9,680 km
Payload perigee speed: 7.3 km/s

The Low Sub MEO tether sends and receivse payloads to and from the upper Super LEO tether.

Super LEO Tether

The anchor mass is in a circular orbit of radius 9300 km.

Upper Super LEO Tether, 765 km long
Safety Factor 3
Zylon taper ratio: 1.4
Tether to payload mass ratio: .84
Tether top radius 10,065 km
Tether top speed: 7.1 km/s
Tether top net acceleration: .11 m/s2 (.011 g)
Payload apogee: 17375 km
Payload apogee speed: 4.1 km/s

The payload apogee is at lunar altitude and the payload's moving .53 km/s. The moon moves at about 1 km/s. So Vinf with regard to the moon is about .47.

Lower Super LEO tether, 450 km long
Safety Factor 3
Zylon taper ratio: 1.13
Tether to payload mass ratio: .29
Tether foot distance from earth 8,844 km
Tether foot speed: 6.2 km/s
Tether foot net acceleration: .7 m/s2 (.07 g)
Payload perigee: 6,778 km
Payload perigee speed: 8.3 km/s

Perigee altitude is about 300 km. Circular orbital speed at this atltitude is about 7.7 km/s. To send a LEO payload on it's way to the Super LEO tether would take about .6 km/s.

Sending a payload from the tether to LEO can take less than .6 km/s as the delta v needed for circularizing can be provided by aerobraking.

Total Tether Mass to Payload Ratio

We've looked at a total of 6 tether lengths, the upper and lower parts of 3 vertical tethers.

Tether Mass to Payload Mass Ratios & Lengths

Length (km)
Upper Super GEO
Lower Super GEO
Upper Sub MEO
Lower Sub MEO
Upper Super LEO
Lower Super LEO

Thus 38 tonnes of Zylon could accommodate 10 tonnes of payload. That's not too bad.

A much larger problem is the anchor mass needed for each tether. There are lots of dead sats just above GEO that could be gathered for the Super GEO tether anchor mass. But anchor masses for the sub MEO and super LEO tethers will be more expensive. This is a possible show stopper.

Facilitating Momentum Exchange

Using Hall Thrusters to restore momentum.

Sending mass from LEO to a lunar height apogee saps our tethers' orbital momentum. The momentum hit is somewhere around payload mass * 4 km/s. Orbital momentum can be restored gradually with ion thrusters. Hall Thrusters can expel xenon with a 30 km/s exhaust velocity.

Plugging these numbers into the rocket equation:

Propellent mass fraction = 1 - e -4/30 = ~.125.

About 1/8. So to make up for the momentum lost throwing 7 tonnes of payload, we'd need a tonne of xenon. Better than chemical but still expensive.

Lunar or NEA propellent as a source of up momentum.

Some Near Earth Asteroids (NEAs) can be parked in lunar orbit for as little as .2 km/s. Carbonaceous asteroids can be up to 40% water by mass (in the form of hydrated clays). There may be rich water ice deposits in the lunar cold traps. So far as I know, these are the most accessible potential sources of extra terrestrial propellent.

Catching propellent from higher orbits would boost a tether's momentum. Dropping this payload to a lower tether would also boost momentum.

Thus up momentum can be traded for down momentum. Xenon reaction mass to maintain tether orbits can be cut drastically with two way traffic.

Jon Goff's gear ratios

Jon Goff has pointed out it take some delta V to get propellent from the moon's surface to LEO. Thus only ~10% of propellent mined lunar cold traps would make it LEO. See his blog post The Slings And Arrows of Outrageous Lunar Transportation Schemes Part-1 Gear ratios.

Well, lunar propellent could be a source of down momentum for the Lunar Sky Hook I described recently. And a source of up momentum for the Trans Cislunar Railroad this blog post looks at. NEA propellent could also be a source of up momentum for the Trans Cislunar Railroad.

Using propellent as a source of tether up momentum I believe it's plausible for 40% of the lunar propellent to make it to LEO. In which case it becomes plausible to use reaction mass to mitigate the extreme conditions of re-entry.

Breaking the Genie's Bottle

The human race is a genie in a bottle. Given Tsiolkovsky's rocket equation, it's enormously difficult to cross the boundaries that confine us. But given infrastructure and resources at our disposal, we can build bridges to larger frontiers.

Monday, August 8, 2016

Lunar Sky Hook

Kim Holder has been urging me to do this blog post. Her comments in various forums have been helpful in thinking about this.

Vertical Lunar Tether In A Polar Orbit

This sky hook is a gravity gradient stabilized vertical tether. It's in a polar orbit so it will pass over the poles as well as the lower lunar latitudes.

Unlike an equatorial orbit, there are only two occasions during a lunar orbit where a tether's Vinf velocity vector is anti-parallel to the moon's velocity vector. So launch windows to earth would only occur each two weeks. That's still pretty often. These occasions are also good times to rendezvous with the tether.

Playing with earth moon three body simulations, polar orbits seem to remain stable up to a radius of around 20,000 kilometers. That is where I will set the anchor mass at the balance point of this sky hook. I believe this is far enough above the lunar surface that the mascons won't damage this tether's orbit.

Asteroid Anchor Mass Via a Keck vehicle

What to use for the anchor mass? With the asteroid retrieval vehicle proposed in the Keck Report, it is possible for a vehicle of moderate mass to retrieve a much larger mass to the earth moon neighborhood. The Keck authors believe a rock could be placed in high lunar orbit for around .17 km/s. A lunar orbit with a 20,000 km radius has a speed of around .5 km/s. I believe it would take around .7 km/s to park a rock in the orbit we want.

The Keck vehicle includes solar panel arrays and Hall ion thrusters. These would be great to have on a vertical tether. It takes awhile for ion engines to impart momentum, but given time they're about ten times as efficient as the best chemical rockets. A tether can build up momentum over time but release it suddenly. Thus they are a good way to enjoy an ion engine's great ISP and an Oberth benefit.

As well as adjusting the tether's orbit the Keck vehicle's solar arrays might also power elevator cars moving up and down the tether. If water is exported from from the lunar cold traps to the tether, the arrays might also crack water into oxygen and hydrogen bipropellent. There are a number of possible uses for this power source.

Upper Tether

The tether length above the anchor mass can be built in increments. I imagine the tether growing longer and more able with time. Here are three possible stages:

To EML2 or EML1

EML1 and 2 are about 65,000 km from the moon. To reach this apolune, we'd need an upper tether length of about 2700 kilometers. Using Wolfe's spread sheet, this tether length has a taper ratio of 1. With a safety factor to 3, tether mass to payload ratio is about .02.

This is pretty good. I believe this low stress tether length could accommodate copper wires to transmit power to the elevator cars.

Once at apolune, I believe it would take about .3 km/s to park the payload at EML2 or EML1.

EML2 is a good staging location should we want to travel to and from destinations beyond the earth-moon neighborhood.

To a Perigee at Geosynchronous Orbit 

Transfer orbit from GEO to the moon is about a an ~36,000 x 378,000 ellipse. Apogee speed is about .45 km/s. The moon's speed is about 1.02 km/s. So the tether needs to hurl a payload to a Vinf of (1.02-.45) km/s or about .57 km/s.

To achieve this Vinf our tether needs to be 12,200 km. Zylon taper ratio is 1.09. With a safety factor of three, Tether to payload mass ratio is about .167. So a ten tonne tether could accommodate a sixty tonne payload. This is still pretty good. A power cable along this length is also doable.

Perigee velocity of our transfer orbit is ~4.13 km/s. Geosynch orbit velocity is ~3.07 km/s. If the transfer orbit and destination geosynch orbit are coplanar, geosynch circularization would be about 1.06 km/s. But I expect that would be the exception rather than the rule. If the orbit inclinations differ by 20º, 1.6 km/s would be needed to park in geosynch.

To a Perigee at Low Earth Orbit.

A 300 x 378,000 km orbit has apogee velocity of ~.19 km/s. (1.02 - .19) km/s = .83 km/s.

To throw a payload to a trans earth orbit, our tether needs to impart a Vinf of .83 km/s. This takes a tether length of 19,200 kilometers. With a safety factor of three, Zylon taper ratio is 1.2. Tether to payload mass ratio is .38.

If perigee is through earth's upper atmosphere, aerobraking can provide a large part of the 3.1 km/s delta V for circularizing at LEO.

Lower Tether

Again, the tether length below the anchor mass can be built in increments. Incremental growth with time is more doable than trying to do the whole length in fell swoop. Here are some possible steps along the way.

To a Perilune at Low Lunar Orbit.

To drop a payload to a 90 km altitude perilune, length needs to be 7360 km. Given a safety factor of 3, Zylon taper ratio is 1.06. Tether to payload mass ratio is .15.

Velocity of transfer orbit's perilune is about 2.2 km/s. Low lunar orbit is about 1.6 km/s. It'd take about .6 km/s to circularize at low lunar orbit. 

To the Moon's Surface, Impact Velocity 1 km/s.

If the tether is extended to a length of 17890 km, tether foot altitude is about 370 km. Dropping a payload from this tether foot would result in a 1 km/s impact. 

Given a safety factor of two, Zylon taper ratio is 2.88. Tether to payload mass ratio is 26.87.

Note the safety factor is less than in the other scenarios. As we descend further into the moon's gravity well, stress climbs more rapidly. It would be more difficult to include copper wires for power along the lower parts of the tether.

To a Tether Foot Just Above the Moon's Surface.

Dropping the tether foot to an altitude of 10 kilometers gives us a length of 18,252 km. Safety factor of 2 and Zylon taper ratio is 3.72. Tether to payload mass ratio is  about 51.

Dropping from this tether foot, a payload would impact the lunar surface at .184 km/s. 

A .2 km/s payload delta V budget for soft landing seems quite doable. Likewise it would take about .2 km/s to launch a payload from the lunar to rendezvous with the tether foot.

However dropping the tether foot this far is considerably more ambitious than the other scenarios described above.

Travel About The Moon

Kim Holder noted such a tether might serve as transportation between locations on the moon.

Without a tether, going from pole to pole would take about 3.4 km/s: 1.7 km/s to launch and another 1.7 for soft landing. Going from equator to pole would take 1.53 km/s to launch and another 1.53 km/s for a soft landing, totaling 3.06 km/s. 

So a 18,000 km lower lunar tether length would make travel about the moon easier.

A Location to Process Asteroid Ore

It takes about .6 km/s to park ore from some of the more accessible asteroids in 20,000 km lunar orbit. If rendezvous with the tether top is doable, it could take considerably less.

I envision infrastructure accreting about the tether anchor mass 18,262 km above the lunar surface. Water, platinum, gold, rare earth metals, and other materials could be extracted at the anchor. Refined commodities could climb to the top of the tether and then tossed earthward.

A Synergy Between The Moon and Near Earth Asteroids

Moon and asteroid enthusiasts are often at odds with one another. They should be allies. In terms of delta V, it's a lot easier to park asteroids in lunar orbit than lower earth orbits. And given growing infrastructure in lunar orbit, the moon's surface becomes more accessible.

Friday, July 22, 2016

Hildas As Cyclers

Hilda Asteroids - Red,   Sun Jupiter Trojans - Blue,   Main Belt - Green

The above image was made from screen captures of Scott Manley's beautiful animation Asteroids In Resonance With Jupiter.

Jupiter is the dot off to the left, the sun is the yellow dot in the middle. Within the Main Belt can be seen Mercury, Venus, Earth and Mars. I colored the different asteroids populations so we can tell them apart.

The Sun Jupiter Trojans have a 1 to 1 resonance with Jupiter. They co-rotate with Jupiter. The leading Trojans remain in a neighborhood 60 degrees ahead of Jupiter and the trailing Trojans stay in a neighborhood 60 degrees behind.

The Hildas have have 3 to 2 resonance with Jupiter meaning they circle the sun three times for every two Jupiter orbits. Jupiter's orbital period is about 12 years and the Hildas have 8 years periods.

The Hilda orbits only look triangular in Manley's animation because they're being viewed in a rotating frame. You can see Jupiter remains on the left side of the image. In an inertial frame, a Hilda orbit is an ordinary elliptical orbit with aphelion passing through the Trojans and perihelion passing through the main belt.

I envision the Hilda biomes playing a similar role as Marco Polo's caravans shuttling people and goods between east and west. But the Hildas travel between the Trojans and the Main Belt.

The would be a series of regular fly bys for a Hilda Cycler

1) Main Belt to trailing Trojans — 4 years.
2) Trailing Trojans to Main Belt — 4 years.
3) Main Belt to leading Trojans — 4 years
4) Leading Trojans to Main Belt — 4 years
5) Main Belt to Sun Jupiter L3 — 4 years. But there is no asteroid population at SJL3.
6) From SJL3 to Main Belt 4 years

Then back to step 1). The cycle repeats itself.

So not only can a Hilda be a go between between the Main Belt and Trojans, but it can also move stuff between the trailing and leading Trojan populations. Trailing to leading takes 8 years and leading to trailing takes 16 years.

As can be seen from Manley's animation, there is a steady stream of Hildas traveling the circuit. 

Delta V

The Hildas have a variety of eccentricities. I will look at a Hilda orbit having an eccentricity of .31. That would put the aphelion at 5.2 A.U. and the perihelion at 2.74 A.U. (The perihelion is in Ceres' neighborhood, Ceres' semi-major axis is 2.77 A.U.).

Assuming a circular, coplanar orbit at 2.74 A.U.,  it would take 2.6 km/s to leave a Main Belt Asteroid and board a Hilda.

Assuming a circular, coplanar orbit at 5.2 A.U., it would take 2.2 km/s to depart the Hilda and rendezvous with a Trojan.

However, coplanar orbits is a very optimistic assumption. Asteroids have a large variety of inclinations. Making a 10 degree plane change from a Hilda's orbit can cost 2 to 3 km/s.

Ways to mitigate delta V expense

Many asteroids spin about pretty fast. This plus their shallow gravity wells make them amenable to bean stalks, also known as space elevators. 

"Why would an asteroid need a space elevator?" I'm sometimes asked. The questioner will assert "It's very easy to get off an asteroid's surface, and getting off the body's surface is the only reason for an elevator." Which is wrong, of course.

Speed of a body on an elevator is ωr where ω is angular velocity in radians per time unit and r is distance from center of rotation. If r is large, the elevator can fling a payload at high velocity with regard to the asteroid. It is quite plausible for an asteroid's bean stalk to provide .5 to 1 km/s delta V.

Also an asteroid bean stalk allows rendezvous with an ion propelled space craft. Ion ships have great ISP but minute thrust. Soft landings with an ion craft are not possible on larger asteroids like Ceres, or Vesta.

And ion propelled ships are more viable in the outer system. When a ship's acceleration is a large fraction of the local gravity acceleration, an ion burn is more like a chemical impulsive burn. See General Guidelines for Modeling a Low Thrust Ion Spiral. In the outer Main Belt, the sun's gravity is about 1 millimeter/sec2. Sun's gravity at the Trojans is about .2 millimeters/sec2.

Jupiter's Trojans

Not much is known about Jupiter's Trojans. Japan hopes to launch a mission in the early 2020's. Another proposed mission is Lucy which would also launch in the early 2020's. 

These bodies are on average 5.2 A.U. from the sun and so receive only 1/27 the sunlight earth enjoys. For this reason I am hopeful they are rich in volatile ices. I'd give better than even odds they have lots of water and carbon dioxide ice. Nitrogen compounds like ammonia and cyano compounds are a possibility. Aside from earth, Nitrogen is in short supply throughout the inner solar system and these would be a great export to the Main Belt biomes.

Their numbers are speculation. According to Wikipedia:

Estimates of the total number of Jupiter trojans are based on deep surveys of limited areas of the sky.[1] The L4 swarm is believed to hold between 160–240,000 asteroids with diameters larger than 2 km and about 600,000 with diameters larger than 1 km. If the L5 swarm contains a comparable number of objects, there are more than 1 million Jupiter trojans 1 km in size or larger. For the objects brighter than absolute magnitude 9.0 the population is probably complete. These numbers are similar to that of comparable asteroids in the asteroid belt. The total mass of the Jupiter trojans is estimated at 0.0001 of the mass of Earth or one-fifth of the mass of the asteroid belt.
Two more recent studies indicate, however, that the above numbers may overestimate the number of Jupiter trojans by several-fold. This overestimate is caused by (1) the assumption that all Jupiter trojans have a low albedo of about 0.04, whereas small bodies may actually have an average albedo as high as 0.12;[16] (2) an incorrect assumption about the distribution of Jupiter trojans in the sky. According to the new estimates, the total number of Jupiter trojans with a diameter larger than 2 km is 6.3 ± 1.0×104 and 3.4 ± 0.5×104 in the L4 and L5swarms, respectively. These numbers would be reduced by a factor of 2 if small Jupiter trojans are more reflective than large ones.[16]
The number of Jupiter trojans observed in the L4 swarm is slightly larger than that observed in L5. However, because the brightest Jupiter trojans show little variation in numbers between the two populations, this disparity is probably due to observational bias. However, some models indicate that the L4 swarm may be slightly more stable than the L5 swarm. 
The largest Jupiter trojan is 624 Hektor, which has an average diameter of 203 ± 3.6 km. There are few large Jupiter trojans in comparison to the overall population. With decreasing size, the number of Jupiter trojans grows very quickly down to 84 km, much more so than in the asteroid belt. A diameter of 84 km corresponds to an absolute magnitude of 9.5, assuming an albedo of 0.04. Within the 4.4–40 km range the Jupiter trojans' size distribution resembles that of the main-belt asteroids. An absence of data means that nothing is known about the masses of the smaller Jupiter trojans. The size distribution suggests that the smaller Trojans are the products of collisions by larger Jupiter trojans.

I'd love to see science fiction stores set on 624 Hektor.

This article written in memory of Hilda Alvarez May 5, 1929 - July 20, 2016

Thursday, May 26, 2016

Rotovator help with re-entry

There are proposals for rotovators to catch payloads in low earth orbit and then throw them to higher orbits.

It occurs to me that rotovators used for throwing comm sats to GTO could also help the upper stage re-enter earth's atmosphere at a lower velocity.

1) Rotovator catches upper stage and payload in LEO.
2) Rotovator throws payload to higher orbit.
3) Rotovator drops upper stage into a suborbital orbit.

Step 3) accomplishes two things:

a) It restores some of the orbital momentum the tether lost in catching and tossing the payload.
b) It reduces re-entry velocity of the upper stage. Slower re-entry velocities make recovery and reuse of the upper stage less difficult.

Is this a good idea? Or another hare brained scheme? I'm tossing this out in several venues hoping knowledgeable folks will review it.

Saturday, April 16, 2016

Liftport Lunar Tether

This is the fifth in a series of posts using Chris Wolfe's spreadsheet to look at various elevators.

274,000 km Lunar Tether

This is based on the Ladder PDF written by Liftport  founder Michael Laine and Marshall Eubanks.

Eubanks and Laine suggest the use of Zylon or M5. This is why I've been using Zylon through out these tether posts. These gentlemen have invested a lot of time and effort researching elevators and tethers. If they like Zylon, I'll follow suit.

They propose launching the tether to EML1. From EML1, the tether anchor would descend moonward towards Sinus Medii on the lunar surface, 0º, 0º. The spent upper stage would drop with the tether foot earthward.

If the mass were tethers alone, the 264,000 length would be inadequate to keep the tether from collapsing to the moon. But spent upper stage acts as a counterweight to maintain tension.

Ratios earthside of EML1

A spent Centaur upper stage is about 2250 kilograms. This is the quantity I used for foot station mass. These newtons subtract from newtons available for payload. The Ladder PDF calls for 11 tonnes of Zylon. By trial and error I entered payload quantities until tether mass in my spreadsheet came to 11 tonnes.

In addition to foot station mass of 2250 kg, I got a maximum foot payload mass of 1640 kg.

Zylon taper ratio: 1.61. Tether mass to payload mass ratio: 8.05

Given the extreme the extreme length of this elevator, I expected a higher number than 8. But the net acceleration at the tether foot is only .0274 newtons per kilogram. With this acceleration, a 10 tonne mass would exert as much force as when my 62 pound dog sits on my lap.

Ratios moonside of EML1

But what sort of payload can this elevator support moonside of EML1?

At the anchor in Sinus Medii, my tether model's cross sectional area is 1.64e-8 square meters. Multipying this times Zylon's tensile strength gives ~95.4 newtons the tether can support. Net acceleration at this point is 1.4 meters/s^2 (mostly moon's gravity). 95.4 newtons/(1.4 m/s^2) = 68 kilograms. For a payload just above the moon's surface, the elevator can support 68 kilograms.

Tether to payload mass ratio: 161.

Let's say we wanted a 1 tonne elevator car capable of carrying 9 tonnes of cargo. We'd need a 1,610 tonne tether.


Dropping a payload from 70,900 km earthward of EML1 would send a payload to to an atmosphere grazing orbit. Repeated perigee aerobraking passes could circularize the orbit. Shedding 3 km/s via repeated drag passes would require some thermal protection but not as much as the space shuttle which would shed 8 km/s over a very short time.

Thus lunar materials could be delivered to Low Earth Orbit (LEO) without using reaction mass.

Likewise, a 3 km/s LEO burn could deliver payloads to an apogee where orbit velocity matches tether velocity. Normal delta V from LEO to moon surface is about 6 km/s. So the elevator cuts about 3 km/s from the delta V budget for reaching the moon's surface. Cutting 3 km/s from delta V budget about doubles payload mass if using H/Lox bi-propellent.

Dropping a payload 160,000 km earth of EML1 would send a payload to an orbit with perigee as geosynchronous orbit altitude. At perigee the circularization burn is .95 km/s. Thus delta V between GSO and lunar surface is less than kilometer per second.

Some drawbacks

This is a very long tether. How fast can an elevator car move? Having copper wire along the length of the tether would boost taper ratio as well tether to payload mass ratio. For descent from EML1 to lunar surface, the tether to payload mass ratio is already 161.

So in addition to carrying gripping wheels and a motor, the elevator car must carry it's own power source. Photovoltaic arrays? There are solar powered golf carts. These aren't famous for their speed. There are Tesla cars whose lithium batteries can be charged by solar cells. These vehicles can move. It is also possible lithium batteries could be charged during an elevator cars down hill descent via regenerative braking. Downhill would be moonward or earthward from EML1. Movement towards EML1 would be uphill.

Batteries, solar arrays and/or regenerative brakes would boost elevator car mass and thus subtract from cargo mass.

Let's say the elevator car can move an average speed of 400 mph (644 kilometers/hour). A round trip along the length of this elevator and back would take about a month. If the elevator doubles payload mass delivered from LEO, it'd take about 160 months to recoup the investment of delivering tether mass from LEO.

And what justifies this investment? What are the benefits of a facility at Sinus Medii?

I'm a moon guy but it's the lunar poles I like. There are polar plateaus that enjoy near constant sunlight and very mild temperature swings. These plateaus neighbor permanently shadowed crater floors that might harbor rich volatile deposits. In situ CHON not only makes life support easier, but extra-terrestrial propellent could break the exponent in the rocket equation.

But Sinus Medii is at the equator. It's as far from the lunar poles as a lunar surface point can possibly be. We're stuck with two week nights, severe temperature swings and regolith drier than a bone.

Charles Radley has suggested mining He3.  I'm not holding my breath but what if we achieved fusion power? Here is John Schilling's take on fusion and lunar He3:
Helium-3 mining on the moon simply does not pass the arithmetic test. The highest 3He concentration ever recorded in lunar regolith is fifteen parts per billion, and the process by which it is deposited is inherently resistant to geologic concentration.
Assuming someone manages to invent a 3He fusion reactor that operates at 50% efficiency (giggle), that translates to net energy output of 4.5E6 joules per kilogram of high-grade regolith.
The energy output of a kilogram of the lowest grade of coal burned in a good 19th-century reciprocating steam engine, is about 4.5E6 joules per kilogram. And that doesn’t change if you substitute dried peat for the coal.
So, the proposal is to set up an enormous mining infrastructure on the Moon, and invent a fundamentally new kind of engine backed by fifty years of failed promises, for the sake of an energy source roughly as good as burning high-grade dirt in a type of engine obsolete for over a century.
And no, that analysis doesn’t change significantly if we include accessible reserves or environmental impact.
I understand that you want desperately to believe that there are immense riches to be had in space, as soon as the suits see the light and come up with the money. The good news is, this is probably true. But the list of great riches to be had in space, does not include lunar helium-3 (or helium-4, for that matter). The numbers do not add up, no matter what the glossy magazine articles say, and math trumps faith.

Other than fuel for fusion it is hard to imagine He3 markets that would justify the expense of a lunar tether and mine.

I admire Michael Laine. I believe tethers will play a part in making space transportation economical. I also like and admire Charles Radley as well as Marshall Eubanks. So it pains me to say this. At this point I am not enthusiastic about the Liftport Lunar elevator.

But there are other possible elevators in the moon's neighborhood.

Thursday, February 11, 2016

Limits to growth, logistic vs exponential

Malthusian growth model

The Malthusian growth model sees population growth as exponential.

P(t) = Poert
P=  P(0) is the initial population size,
r = population growth rate
t = time

Growth of microbe populations are often used to illustrate this. Let's say an amoeba will grow and divide into two amoeba after an day of absorbing nutrients.

Day 1: 1 amoeba
Day 2: 2 amoeba
Day 3: 4 amoeba
Day 4: 8 amoeba

And so on. Population doubles each day. Exponential growth is famous for starting out slow and then zooming through the roof.

On the left is exponential growth in cartesian coordinates. On the right in polar coordinates, radius doubles every circuit.

Malthus imagined a rapidly growing population consuming all their available food supply and then starving to death.

Logistic growth

Sometimes populations have suffered Malthusian disaster. More often rate of growth slows as the population approaches the limit that resources can support. This is logistic growth.

P(t) = Le-rt / (L +( e-rt - 1))

Where L is the maximum population local resources can support.

At the start, logistic growth resembles exponential growth. But as the population nears the logistic ceiling, growth tapers off. Above the blue boundary represents the limit to growth. In red is the logistic growth curve, the thinner black curve is exponential growth.

What slows growth?

In Heinlein's science fiction, war limits growth. This was also the foundation idea of Niven and Pournelle's The Mote In God's Eye -- War is the inevitable result of burgeoning populations.

The Four Horsemen of Apocalypse -- plague, war, famine and death are seen as natural outcomes of uncontrolled population growth.

A declining fertility rate is a less ominous way to step on the brakes. It is my hope most people will choose to have small families. And indeed, current trends indicate people are voluntarily having fewer kids. Still, there are skirmishes as various entities compete for limited resources.

Bad vs worse

A growing population, a growing consumer appetite, a limited body of resources. It doesn't take a rocket scientist to see growth must eventually level off.

Whether it levels off via the 4 horsemen or moderation and voluntary birth control, either option sucks.  It's disaster vs stagnation.


Above is a Johnny Robinson cartoon. Used with permission.

I believe our solar system is possibly the next frontier. That has been the thrust of this blog since the start. If we do manage to break our chains to earth, it will be a huge turning point in human history, more dramatic than the settling of the Americas. The potential resources and real estate dwarf the north and south American land masses.

While settling the solar system allows expansion, it won't relieve population pressure on earth. Settlement of the Americas did not relieve population pressure in Europe, Asia and Africa. Mass emigration is impractical.

Rather, pioneers jumping boundaries starts growth within the new frontiers. I like to view the logistic growth spiral in polar form as a petri dish. When a population within a petri dish has matured to fill its boundaries, it sends spores out to neighboring petri dishes. Then populations within neighboring petri dishes grow to their limits.

The first petri dish still has a population filling the limit. They have not escaped the need to live within their means. I take issues with critics who say space enthusiasts want to escape to a new planet after earth has been trashed. Space enthusiasts know earth is fragile, more so than the average person. It is noteworthy that Elon Musk is pioneering planet preserving technologies such as electric cars and solar energy.

But even if mass emigration from Europe or Asia was not possible, the expansion into the Americas energized the economy and zeitgeist of the entire planet. It provided investment opportunities. Also an incentive to explore. This is the greatest benefit of a frontier. Curiosity is one of the noblest human qualities and I hope we will always want to see what lies over yonder hill. And that we will keep devising ways to reach the far side of the next hill. Satisfaction and contentment are for cattle. If we lose our hunger and wander lust we will no longer be human.

Sunday, January 17, 2016

Fact checking Neil deGrasse Tyson

Tyson is well known for fact checking movies, comics and other pop culture stuff. Here's giving Tyson a taste of his own medicine.

Stories of President Bush and Al Ghazali

In a number of vids Tyson talks about the Golden Age of Islam and Bush's response to the 9-11 terrorists. George Bush and Star Names and Naming Rights are a couple.

George Bush

The videos start with Tyson describing a speech Bush gave shortly after the 9-11 attack. According to Tyson, Bush quotes the bible saying God named the stars in order to "distinguish we from they." "They" being Muslims. Stands to reason right? We all know a Republican would seize this emotionally charged moment to stir up hatred against Arabs.

Except that Bush never gave this post 9-11 speech. His actual 9-11 speech called Islam the religion of peace. Bush was calling for inclusion and tolerance. Exactly the opposite of the xenophobic demagogue Tyson falsely portrays.

Bush did quote scripture in his eulogy for the Space Shuttle Columbia astronauts. In no way was this eulogy a xenophobic slam against Arabs.

Hamid Al-Ghazali

In the naming rights videos Tyson likes to blame Hamid Al-Ghazali for the end of the "Islamic Golden Age". From 800 to 1100, Islamic civilization made great strides in mathematics and science. Much of the star names we use today are Arabic due to the work of Arabic astronomers in that era. According to Tyson Al-Ghazali wrote mathematics was the work of the devil.

Did Al-Ghazali demonize mathematics? So far as I know, Tyson hasn't given any cites supporting this notion. But numerous historians dispute this claim. Here's a Reddit AskHistorian Thread, a blog post by Yusuf Chaudhary, and a Reddit BadHistory thread.

Tyson's name turns up often in the badhistory subreddit group.

Elaboration on Hamid al Ghazali

In the comments below Tyson back pedals from his claim that Ghazali said mathematics was the work of the devil. He writes
As for Al Ghazali, a more accurate representation of his views is that the manipulation of numbers was an earthly rather than a divine pursuit.
I've been reading Ghazali's writings to get some of his statements in context. I've come to the conclusion that Ghazali does indeed discourage his followers from pursuing the discipline of mathematics. Tyson has a point. But it is wrong to say Ghazali characterized math as the work of the devil.

Five Centuries Regressed

Is the earth flat or round? This silly argument between Tyson and rapper B.o.B. generated a great deal of publicity for B.o.B., Tyson, and Tyson's nephew.

Part of the exchange: "@bobati Duude — to be clear: Being five centuries regressed in your reasoning doesn't mean we all can't still like your music."

Supposedly folks during the dark ages thought the earth was flat. Sadly Tyson is perpetuating this myth.

In the August 1991 issue of History Today Jeffrey Russel effectively argues people knew the earth was round during and before the time of Columbus.

In his comment reply (below) Tyson perpetuates the myth that knowledge of a spherical earth was lost in the "Dark Ages". Historian Tim O'Neill explains where this myth comes from. O'Neill also documents prominent scholars from that period that knew the earth was spherical.

The above links as well as more interesting reading can be found in this reddit badhistory thread on Tyson's battle with B.o.B.

A Rainbow Forms Only Broadside
To Your Line Of Sight

Neil seems to be saying the rainbow lies in a plane perpendicular with your line of sight. We'll go with that. But what does Tyson mean by "line of sight"?

A Tyson defender who goes by the name of Peef_Rimgar wrote
"... if you look it up it is defined as the straight, uninterrupted line between your eyes and something you can see. "That building is in my line of sight" doesn't mean it has to be directly right in front of me."

We'll look at Peef's definition. It includes the multitude of lines passing through the viewer's pupil and striking the retina.

Lines of sight from the edge of the rainbow to the viewers eye intersect the rainbow plane at 48º. Not perpendicular.

There is one line through the viewer's pupil that is perpendicular to the plane. The line that passes through the sun as well as the viewer's head. Also lying on this line is the shadow of the viewer's head. The shadow of the head occupies the rainbow's center. I'll call this line the center line.

Now the viewer turns his head to the right. The rainbow remains perpendicular to the center line. However now the center line lies outside of the viewer's cone of vision. In this case there are zero lines of sight perpendicular to the plane.

The rainbow does remain perpendicular to the line passing through the sun, the viewer's head and the shadow of the viewer's head. To reach the Pot of Gold, the viewer would have to detach himself from his shadow and walk to the rainbow's end. Clearly impossible. But calling this line "line of sight" is a sloppy, inaccurate label.

Blind As A Bat

This common misconception is addressed in  Christie Wilcox's Discover article Actually, Bats See Just Fine, Neil.

Tyson's trailer for The Martian

Hermes' impossible trajectory

Above is a link to Neil deGrasse Tyson's trailer for The Martian. At 1:15 of the vid, Tyson has the space ship Hermes departing from Low Earth Orbit (LEO). 124 days later he has Hermes arriving at Mars orbit (2:17 of the video).

Hermes is propelled with low thrust ion engines. In the book when Hermes is about to rendezvous with Watney's Mars Ascent Vehicle (MAV), Lewis says Hermes can do up to 2 mm/s2. This acceleration is also given online:

Two millimeters per second squared would require an extremely good alpha. But it's possible future power sources will deliver more watts per kilogram. So 2 mm/s2 is only medium implausible. I'll let this slide.

Problem is, low thrust ion engines really suck at climbing in and out of planetary gravity wells. From low earth orbit, it would take Hermes about 40 days to spiral out of earth's gravity well and about 20 days to spiral from the edge of Mars' gravity well to low Mars orbit. Two months spent climbing in and out of gravity wells destroys Andy Weirs' 124 day trajectory.

Given 2 mm/s2, the trajectory Tyson describes is flat out impossible.

A slow ride through the Van Allen belts.

At 1:50 of Tyson's video he talks about the danger of solar flares and how astronauts are vulnerable to radiation. Well, departing from LEO means a month long spiral through the Van Allen Belts. Not only does the long spiral wreck Weir's 124 day trajectory, it also cooks the astronauts.

Tyson enjoys some notoriety for fact checking fantasies like Star Wars or The Good Dinosaur. This leaves me scratching my head. Many of the shows he fact checks make no pretense at being scientifically accurate. However The Martian was an effort at scientifically plausible hard science fiction and thus is fair game. Same goes for Tyson's trailer.

A physically impossible trajectory along with cooking the astronauts? Tyson's effort at hard science fiction isn't any better than Gravity or Interstellar.

Neil's Five Points of Lagrange Essay

The Five Points of Lagrange was a Neil deGrasse Tyson article published in the April, 2002 issue of Natural History Magazine.

A few excerpts:

Popular usage has made "exponential" a general term for dramatic change. But a physicist should know the more specific mathematical meaning of the this word. Gravity falls with inverse square of distance, not exponentially.

Wrong. Clarke's contribution was suggesting communication satellites be placed in geosynchronous orbit (GSO). A fantastic idea with tremendous impact. But Clarke wasn't the first to calculate the altitude of GSOs.

Herman Potočnik calculated the altitude of GSO in 1928.  It's possible this altitude was calculated even earlier. Newton might have done it.

Unhackable Systems

The solution is so simple, just make unhackable systems. Oh my gosh, why didn't the cyber security folks ever think of that?

Twitchy published some good responses.

Tyson on "idiot doctors"

The first half of the video Tyson argues it's foolish to believe surviving cancer demonstrates divine intervention. I'm fine with that.

But the second half of the video is a clueless rant against idiot doctors, the American Medical Association and Pre-Med students.

So a patient lives longer than predicted. Does this make the doctor an idiot? No. Typically a doctor will give his patient statistics for people in a similar condition. If someone lives longer than the norm, it demonstrates there are statistical outliers on a bell curve. It is..... astonishing. Astonishing that Tyson and the physics 101 prof are unfamiliar with entry level statistics.

Also Tyson as well as the physics prof seem to believe someone who's failed freshman physics would go on to med school. There are idiot physicists, I assure you!

A more thorough fisking of this vid can be found at the Tracinski Letter. Robert Tracinski is an atheist, by the way.

The Coriolis Force

The Coriolis Force was a Tyson article published in the March 1995 issue of Natural History. In the article Neil has this to say about the 1914 Falklands battle:
But in 1914, from the annals of embarrassing military moments, there was a World War I naval battle between the English and the Germans near the Falklands Islands off Argentina (52 degrees south latitude). The English battle cruisers Invincible and Inflexible engaged the German war ships Gneisenau and Scharnhorst at a range of nearly ten miles. Among other gunnery problems encountered, the English forgot to reverse the direction of their Coriolis correction. Their tables had been calculated for northern hemisphere projectiles, so they missed their targets by even more than if no correction had been applied. They ultimately won the battle against the Germans with about sixty direct hits, but it was not before over a thousand missile shells had fallen in the ocean.
However the role of Coriolis correction in this battle is a an urban legend.

No Northern Penguins
To which Sean Davis replies noting there are Galapagos Penguins.

Brick Helicopters

Helicopter blades will continue rotating after engine failure. Descending through the air at an angle can spin up the blades. Leveling off just before reaching the ground makes for a soft landing.

The process is described and demonstrated at this Getting Smarter Every Day Video.

GMO = artificial selection

In this video Tyson defends genetic modification by claiming it's not different from the artificial selection humans have been practicing for millennia.

Which is wrong. Genetic modification as practiced by Monsanto is splicing DNA from one species onto the DNA of another species. Artificial selection encourages traits that already exist in a population's gene pool. Here is a primer: Genetic Modification Explained.

Are GMOs beneficial? Or are they harmful? I don't know. Here I'm not taking a position pro or con GMOs. I'm pointing out Tyson's argument conflates two different techniques.

Are there more Tyson bloopers?

I've only seen a fraction of Tyson's prolific output so I suspect I'm just scratching the surface. If readers know of more Tyson bloopers, drop me a line. The folks at The Federalist are also enthusiastic Tyson fact checkers.