Thursday, September 15, 2016

Xenon

Xenon The Noble Gas

Xenon is one of heavier Noble Gases

Screen capture from ChemicalElements.com

The noble gases are the orange column on the right of the periodic table. These are chemically inert. Which means they're not corrosive. This makes them easier to store or use.

Low Ionization Energy

Per this graph is from Wikipedia, Xenon has a lower ionization energy than the lighter noble gases.


Ionization energy for xenon (Xe) is 1170.4 kJ/mol. Ionization for krypton (Kr) is 1350.8 kJ/mol. Looks like about a 15% difference, right?

But a mole of the most common isotope of xenon is 131.3 grams, while a mole of krypton is 82.8 grams. So it takes 181% or nearly twice as much juice to ionize a gram of krypton.

Likewise it takes nearly 4.5 times as much juice to ionize a gram of argon.

The reaction mass must be ionized before it can be pushed by a magnetic field. Xenon takes less juice to ionize. So more of an ion engine's power source can be devoted to imparting exhaust velocity to reaction mass.

Big Atoms, Molar Weight

Low molar weight makes for good ISP but poor thrust. And pathetic thrust is the Achilles heel of Hall Thrusters and other ion engines. The atomic weight of xenon is 131.29 (see  periodic table at the top of the page).

Tiny hydrogen molecules are notorious for leaking past the tightest seals. Big atoms have a harder time squeezing through tight seals. Big whopper atoms like xenon can be stored more easily.

Around 160 K, xenon is a liquid with a density of about 3 grams per cubic centimeter. In contrast, oxygen is liquid below 90 K and a density of 1.1. So xenon is a much milder cryogen than oxygen and more than double (almost triple) the density.

Abundance

Ordinary atmosphere is 1.2 kg/m3 while xenon is about 5.9 kg/m3 at the same pressure. Xenon has about 4.8 times the density of regular air.

By volume earth's atmosphere is .0000087% xenon. 4.8 * .000000087 = 4.2e-7. Earth's atmosphere is estimated to mass 5e18 kg. By my arithmetic there is about 2e12 kg xenon in earth's atmosphere. In other words, about 2 billion tonnes.

Page 29 of the Keck asteroid retrieval proposal calls for 12.9 tonnes of xenon. Naysayers were aghast: "13 tonnes is almost a third of global xenon production for year! It would cause a shortage." Well, production is determined by demand. With 2 billion tonnes in our atmosphere, 13 tonnes is a drop in the bucket. We throw away a lot of xenon when we liquify oxygen and nitrogen from the atmosphere.

In fact ramping up production of xenon would lead to economies of scale and likely cause prices to drop. TildalWave makes such an argument in this Space Stack Exchange answer to the question "How much does it cost to fill an ion thruster with xenon for a spacecraft propulsion system?" TildalWave argues ramped up production could result in a $250,000 per tonne price. That's about a four fold cut in the going market price of $1.2 million per tonne.

Radon

If you examined the periodic table and ionization tables above you might have noticed there's a heavier noble gas that has an even lower ionization energy: Radon a.k.a. Rn.  Radon is radioactive. Radon 222, the most stable isotope, has a half life of less than 4 days. If I count the zeros on the Radon page correctly, our atmosphere is about 1e-19% radon -- what you'd expect for something with such a short half life. Besides being rare, it wouldn't last long in storage.

Where xenon excels


Great for moving between heliocentric orbits

Ion thrusters can get 10 to 80 km/s exhaust velocity, 30 km/s is a typical exhaust velocity. That's about 7 times as good as hydrogen/oxygen bipropellent which can do 4.4 km/s. But, as mentioned, ion thrust and acceleration are small. It takes a looong burn to get the delta V. To get good acceleration, an ion propelled vehicle needs good alpha. In my opinion, 1 millimeter/second2 is doable with near future power sources.

If the vehicle's acceleration is a healthy fraction of local gravity field, the accelerations resemble the impulsive burns to enter or exit an elliptical transfer orbit. But if the acceleration is a tiny fraction of the local gravity field, the path is a slow spiral.


Earth's distance from the sun, the sun's gravity is around 6 millimeters/second2. At Mars, sun's gravity is about 2.5 mm/s2 and in the asteroid belt 1 mm/s2 or less. Ion engines are okay for moving between heliocentric orbits, especially as you get out as far as Mars and The Main Belt.

Sucks for climbing in and out of planetary gravity wells

At 300 km altitude, Earth's local gravity field is about 9000 millimeters/second2. About 9 thousand times the 1 mm/s2 acceleration a plausible ion vehicle can do. At the altitude of low Mars orbit, gravity is about 3400 millimeters/sec2. So slow gradual spirals rather than elliptical transfer orbits. There's also no Oberth benefit.

At 1 mm/sec2 acceleration, it would take around 7 million seconds (80 days) to climb in or out of earth's gravity well and about 3 million seconds (35 days) for the Mars well.

Mark Adler's rendition of an ion spiral
where the thruster's acceleration is 1/000 that of local gravity at the start.

The general rule of thumb for calculating the delta V needed for low thrust spirals: subtract speed of destination orbit from speed of departure orbit.

Speed of Low Earth Orbit (LEO) is about 7.7 km/s. But you don't have to go to C3 = 0, getting past earth's Hill Sphere suffices. So about 7 km/s to climb from LEO to the edge of earth's gravity well.

It takes about 5.6 km/s to get from earth's 1 A.U. heliocentric orbit to Mars' 1.52 A.U. heliocentric orbit.

Speed of Low Mars Orbit (LMO) is about 3.4 km/s. About 3 km/s from the edge of Mars' Hill Sphere to LMO.

7 + 5.6 + 3 = 15.6. A total of 15.6 km/s to get from LEO to LMO.

With the Oberth benefit it takes about 5.6 km/s to get from LEO to LMO. The Oberth savings is almost 10 km/s.

10 km/s is nothing to sneeze at, even if exhaust velocity is 30 km/s. Climbing all the way up and down planetary gravity wells wth ion engines costs substantial delta V as well as a lot of time.

Elevators and chemical for planet wells, ion for heliocentric

So in my daydreams I imagine infrastructure at the edge of planetary gravity wells. Ports where ion driven driven vehicles arrive and leave as they move about the solar system. Then transportation from the well's edge down the well would be accomplished by chemical as well as orbital elevators.



Other possible sources of ion propellent.

Another possible propellent for ion engines is argon. Also a noble gas. Ionization energy isn't as good as xenon, but not bad. Mars atmosphere is about 2% argon. Mars is next door to The Main Belt. I like to imagine Mars will supply much of the propellent for moving about the Main Belt.

Saturday, September 10, 2016

General template for space elevators

A Family of Conic Sections

Below is a general vertical space elevator. The conic sections are the paths payloads would follow if released from a point on the tether a distance r from body center.

We choose our units so radius of the balance point is 1. Centrifugal acceleration matches gravity at the balance point and net acceleration is zero. For tether locations above the balance point, centrifugal force exceeds gravity and net acceleration is up (away from the planet). For locations below the balance point, gravity is greater than centrifugal acceleration and the net acceleration is down.


This family of conic sections are coplanar, coaxial and confocal. Eccentricity is r3-1, setting r = 1 at the circular orbit of the balance point. (See this stack exchange answer for the math).

In the yellow region are hyperbolic orbits. In the blue region are are elliptical orbits higher than the circular orbit at the balance point. In the orange region, the tether drops payloads into elliptical orbits lower than the circular orbit at the balance point.

A circle of eccentricity zero separates the orange and blue regions, radius of circle = 1.
A parabola of eccentricity 1 separates the blue and gold regions, radius of parabola's periapsis = 21/3

Here is the same graphic zoomed in:


Here is the graphic as a Scalable Vector Graphic. I am hoping science fiction writers and illustrators will download this resource and use it.

Scaling this graphic for a variety of scenarios:


The numbers are in kilometers. In the case of earth, the circular orbit is the geosynchronous orbit at an altitude of about 36,000 kilometers.

In general, radius of a synchronous orbit can be described as:

r = (Gm / ω2)1/3

Where ω is the body's angular velocity in radians, 2 pi radians/sidereal day.

Orbital Elevators

We usually think of an a space elevator anchored at the body's equator. An elevator can also be in a non synchronous orbit. Here the template is scaled to match the orbits of Phobos or Deimos:



Notice Phobos' tether foot  is above Mars surface. The foot is moving about .5 km/s with regard to Mars surface and therefore can't be anchored to Mars. Neither could a Deimos elevator be attached to Mars.

Orbital radius of Phobos is about 40% that of Deimos. So I cloned and shrunk Deimos' tether conics by 40%. I rotated the cloned family of conics by 180º.  The result is an interesting moiré pattern:


It was this pattern that led me to search for a common ellipse.

Eccentricity of the common ellipse:

e = (1 - (ωDeimosPhobos)1/2) / (1 + ωDeimosPhobos)1/2)

Periapsis and apoapsis of the common ellipse:

rperiapsis = (1 + e)1/3 rPhobos
rapoapsis = (1 - e)1/3 rDeimos

ZRVTOs

Here's a pic of the ellipse Phobos and Deimos share:


This is an example of a Zero Relative Velocity Transfer Orbits (ZRVTO) - a term coined by Marshall Eubanks. In Marshall's words: "locations (and times, say for a Lunar and Terrestrial space elevator) where you drop things from one space elevator and they approach and hang motionless (for an instant) at a location on the other elevator.  ... what you would want for large scale movement of material."

Eubanks goes on to say "In practice, you might need a little bit of course correction delta-V to make up for radiation pressure, etc."

Also it would be rare for the elevators playing catch to be perfectly coplanar. So a small plane change delta V expense will be the rule rather than the exception. Still the delta V budgets would be a small fraction of what it would take for normal lift off and insertion to Hohmann transfers.

Not just Phobos and Deimos

To be an anchor for a vertical elevator, a moon needs to be in a near circular orbit and tide locked to its planet. This describes most of the moons in our solar solar system. For two moons to share an ellipse, they need to be nearly coplanar. Again, most the moons in our solar system.

Here are the common ellipses between the moons of Saturn:


Judging by the two gas giants and two ice giants in our solar system, families of coplanar, tidelocked moons are common.

Mini Solar Systems

Earlier I had looked at Mini Solar Systems, a notion I stole from Retrorockets. In our solar system Hohmann trip times between planets are on the order of months or years. Launch windows are typically years apart. But for a system of moons around a gas giant, trip times and launch windows are days or weeks. So a Flash Gordon paced story could take place without wildly improbable engineering.

GIELO and ELM

GIELO - Giant In Earth Like Orbit. ELM - Earth Like Moon. I have long been infatuated with this setting. Here is a painting I had done in 2001:


ELM the earth like moon is in the upper right. In the foreground a generation star ship is sending quad pod scout probes to investigate an artifact at the GIELO-ELM L4 region.

James Cameron's Avatar uses such a setting. Pandora is an ELM. I believe this setting could be developed a lot more. If ELM had sister moons and they were all tide locked, it would be a nice mini-solar system setting.

Icey moons with hospitable interiors.

Gas giants in Goldilocks zones aren't the only possibility. Temperature and pressure rise as we burrow deeper into a body. Earth might not be the only location in the solar system that has liquid water at a livable pressure. Thus the icey moons of our own solar system might eventually become "mini solar systems".

Planets of red dwarfs

And recently an approximately earth sized planet was found in the goldilocks zone of Proxima Centauri. Proxima Centauri is a small red dwarf star. The possibly earth like planet has an orbital radius of about 7.5 million kilometers and an orbital period of about 12 days. Planets about small red dwarfs are yet another possible "mini solar system" setting. Planets so close are likely tide locked to the star. Would atmospheric convection mitigate the temperature extremes between the night side and day side? I'm not sure. In any case, I believe there would be a comfortable region hugging the planet's frozen terminator. (By "frozen" I mean stationary).

Delta V and the rocket equation

The Retrorockets guy took a second look at mini solar systems. While trip times are short and launch windows frequent, it still takes a lot of delta V to insert to a Hohmann transfer. I was annoyed he used the incorrect Tom Murphy method of patching conics. But most his math is sound. He is correct that Tsiolkovsky's rocket equation would be a major pain in the mini solar system just as it is in ours. 

This is where a tether system comes into play. Given elevators on tide locked bodies and assuming most the bodies are nearly coplanar, travel between bodies could be done with very little reaction mass. It'd still take a lot of energy to move stuff up and down the elevators. But the difficult mass fractions imposed by the Tsiolkovsky's equation would no longer be a consideration.

Summary

Similar mathematical models and drawings can be used for a wide range of vertical tethers.

A popular misconception is that elevators are only good for getting off the ground. So it's a waste to build an elevator from a small body. But an elevator not only gets the payload off the ground, it can fling a payload towards a destination. The hyperbolic orbits portrayed in this post are especially interesting.

Space elevators would be especially useful in a system of tide locked moons. Or tide locked planets about a small star.

So far the only elevators I see portrayed in science fiction are from major planets. Like Kim Stanley Robinson's elevators in his Mars trilogy. Or Clarke's earth elevator in Fountains of Paradise. There are far more plausible elevators that could be very useful. These doable elevators could also provide many interesting settings.