Saturday, April 6, 2019

Bridenstine's Why The Moon Matters

Back in December 29, 2016 Bridenstine made a blog post "Why The Moon Matters". Bridenstine was representative of Oklahoma at the time.

Sadly the post was taken down when Bridenstine left his post as representative and became NASA administrator. But I recently found the post using the Wayback Machine.

Bridenstine's reasons were pretty the same arguments made by lunar scientist Paul Spudis (RIP).

I post it here for historical reference. Copying and pasting:

Jim's Blog

Why the Moon Matters

by Rep. Jim Bridenstine

f t # e
Washington, December 29, 2016 0 comments
On July 20, 1969, the free world won the space race when an American flag was planted on the Moon. Twelve Americans walked on the Moon during the Apollo program, resulting in a treasure trove of knowledge not only about the Moon, but about the universe.  Even better, by demonstrating the United States’ political, economic, and technological prowess, it played a part winning the Cold War. In 1983, Ronald Reagan introduced the Strategic Defense Initiative to defend the free world from nuclear ballistic missiles. While many called it destabilizing, and even suggested it was impossible to achieve, the Soviet Union took it very seriously, made every effort to eliminate it, and spent whatever it took to compete. They eventually went bankrupt.  SDI, while not fully implemented, was a geopolitical success built on the technical credibility provided by Apollo. As Ronald Reagan predicted, “We win. They lose.”

Through SDI, the Brilliant Pebbles program was born as a space based system to track and destroy ICBMs. Years later, in 1994, a Brilliant Pebbles satellite was repurposed to orbit and map the Moon. That mission, called Clementine, tested military sensors and made history when it provided evidence of lunar water ice. Later experiments by NASA and other space agencies indicated billions of tons of water ice at each lunar pole.

This single discovery should have immediately transformed America’s space program. Water ice not only represents a critical in situ resource for life support, but it can be cracked into its components, hydrogen and oxygen, to create the same chemical propellant that powers rockets.

All of this is available on a world that has no atmosphere and a gravity well that is 1/6th that of Earth. In other words, standard aerodynamic limitations do not apply, permitting the placement of the propellant into orbit either around the Moon or around the Earth.

From the discovery of water ice on the Moon until this day, the American objective should have been a permanent outpost of rovers and machines, with occasional manned missions for science and maintenance, in order to utilize the materials and energy of the Moon to drive down the costs and increase the capabilities of American operations in cis-lunar and interplanetary space.

Water ice on the Moon could be used to refuel satellites in orbit or perform on-orbit maintenance. Government and commercial satellite operators could save hundreds of millions of dollars by servicing their satellites with resources from the Moon rather than disposing of, and replacing, their expensive investments. Eventually, the customers of Direct TV, Dish Network, internet broadband from space, satellite radio, weather data, and others could see their bills reduced and their service capacities greatly increased.

While most satellites are not currently powered by liquid oxygen and liquid hydrogen, next generation satellite architectures could utilize lunar propellant if low-cost in-orbit servicing were available. Commercial operators will follow if the United States leads with its own constellations.  Such leadership would require a whole-of-government approach with the interagency support of the newly reconstituted National Space Council. The objective is a self-sustaining, cis-lunar economy, whereby government and commercial operators save money and maximize the utilization of space through the use of lunar resources.

This is also the first step for manned missions deeper into our solar system. A permanent human presence on other celestial bodies requires in situ resource utilization. The Moon, with its three-day emergency journey back to Earth, represents the best place to learn, train, and develop the necessary technologies and techniques for in situ resource utilization and an eventual long term human presence on Mars. Fortunately, the Space Launch System and Orion will start testing in 2018. This system, with a commercial lander, could quickly place machines and robots on the Moon to begin the cis-lunar economy. With the right presidential guidance, humans could return in short order as well; this time, to stay.

There are other economic benefits to a permanent presence on the Moon. Utilization of lunar oxides for in situ additive manufacturing (3-D printing) could sustain and develop lunar operations. If economical, we should pioneer the extraction of highly valuable platinum group metals and the ability to transport them back to Earth. The development of practical solar power satellites that beam energy directly to all areas of the Earth is made possible through the use of the resources of the Moon. Research on this concept is already being done in Japan, as well as at the Naval Research Lab here in the United States. The United States government should lead the way in retiring risk for these endeavors with the intent to empower commercial companies to sustain the cis-lunar economy. This could fundamentally alter the economic balance of power on Earth.

As the cis-lunar economy develops, competition for locations and resources on the Moon is inevitable. The Chinese currently have landers and rovers on the Moon. The United States does not. Very soon, the Chinese will be the first of humanity to explore the far side of the Moon and place robots at the poles. As my friend Congressman Bill Posey says, “They are not going there to collect rocks.” China has its own manned space station. The United States’ commitment to the International Space Station ends in 2024. China has a domestic capability to launch its Taikonauts into orbit. The United States relies on Russia. American adversaries are testing antisatellite weapons and proliferating satellite jamming, spoofing, and dazzling technologies. It is time for the United States to re-posture and assert true space leadership.

It must be stated that constitutionally, the U.S. government is required to provide for the common defense. This includes defending American military AND commercial assets in orbit, many of which have the dual role of providing commercial and military capabilities. The same applies for assets on and around the Moon. The U.S. government must establish a legal framework and be prepared to defend private and corporate rights and obligations, all keeping within the 1967 Outer Space Treaty. The United States must have cis-lunar situational awareness, a cis-lunar presence, and eventually must be able to defend freedom of action in space. Cis-lunar development will proceed with American values and the rule of law if the United States leads.

Space utilization has transformed the human condition, including how we communicate, navigate, produce food and energy, conduct banking, predict weather and perform disaster relief. While many of these gains are a result of private investment and commercial markets, they are only possible because the United States government took the lead and retired risk for these capabilities. Today, we are experiencing a space renaissance. The first launch of the Space Launch System is less than two years away. In 2021, we will use the Orion capsule to send astronauts beyond low Earth orbit for the first time since the 1970s. Commercial launch vehicles are maturing and commercial deep space habitats are currently in development. A renewed focus on utilizing the Moon can help further these advances and achievements. The choices we make now can forever make America the preeminent spacefaring nation.



Wednesday, March 13, 2019

Orbital Mechanics Coloring Book 2nd edition

Costs

The folks making the first edition went belly up. So the first edition is out of print. I want to do a second edition. The makers of the first edition would print it in batches of around 50 (I believe) and cost of printing was $6.00/book. Suggested Retail Price was $10.00.

Checking my Amazon author's page over the years I believe around 2000 coloring books were sold. When you get into quantities in the thousands, set up costs of traditional printing are amortized over more units and traditional lithographic printing is cheaper.

I recently asked for quotes:


Given the small cost difference between 56 and 64 pages, I've decided to go for 64 pages.

The first edition was 40 pages. Given the new book has 24 more pages and the cost is a lot less, I believe it's a good gamble I can sell 4000 2nd edition coloring books.

I'm thinking of an Suggested Retail Price of $4.99 and to retailers I would charge $2.99.

Sadly the past decade or two hasn't been kind to the newspaper industry. Our weekly newspaper is still hanging on but my sister and business partner likes to say we're in the buggy whip business. I am thinking of attempting a Kick Starter to fund the printing.

New in the second edition

Given 24 more pages I can add a lot of extra stuff. I've kept most of the original 40 pages and added:

Page 18


In the section on Kepler's 2nd Law I've added a visualization that helps show r X v is twice the area swept out over a given time period. That specific angular momentum is twice the area of the ellipse per orbital period.

Page 22


Page 22 attempts to portray my visualization that helps me remember centrifugal acceleration is ω2r.

Pages 28 and 29



Attempts to explain radians and to show circular motion is ωr where ω is angular velocity in radians.

Pages 30 to 35

Are devoted to orbital vertical tethers. I am going to try to start calling these Sarmount tethers as I have recently learned Eagle Sarmount proposed these in the 1990s.

Perhaps science fiction device but I like them any way. The geometry and math associated with these is pleasing, in my opinion. Here are two pages from this section:



Pages 49 - 51

Are about the Oberth Benefit and EML2



Pages 51 - 52

Are about the rocket equation and mass fractions.



Page 64

Will be resources that have helped me. Books, websites, forums. Atomic Rockets, NasaSpaceflightForums, Space Stack Exchange, Tough SF and others. I am adding to this list as more occur to me.

Pages 53 to 63

Still to be made. Any suggestions?

Front and back cover



Front and back cover will be a couple of my more playful drawings. Text for delta V map to be added in. Also Suggested Retail Price, ISBN number and barcode. The section on Dandelin Spheres remains in the coloring book and the front cover will one of the Dandelin drawings colored in.

The earlier book was labeled a workbook. The printers of the first edition told me this was because ISBN numbers for workbooks are free. However I want this book to be playful as well as informative. In my opinion something categorized as a coloring book is more marketable than a workbook.

Inside front cover

Will be my favorite equations. The Vis Viva Equation will be at the top. I've been thinking of making a reference sheet to pin to the wall next to my computer. This would serve.

Here is the coloring book as of  June, 2019 (6.4 MB pdf, not too big). Reviews would be appreciated. Steven Pietroban invested a fair amount of time looking over the first edition and found many small errors and a few substantial errors. Given my tendency to make misteaks, I'm sure there are errors hiding in my more recent effort. A heads up would be much appreciated if you see something wrong.

My email is hopd at cunews dot info.






Saturday, June 30, 2018

Asteroid Day

On June 30 in 1908 the Tunguska object took out a good chunk of forest in Siberia.

In February of 2016 the United Nations approved a resolution stating

30 June International Asteroid Day to observe each year at the international level the anniversary of the Tunguska impact over Siberia, Russian Federation, on 30 June 1908 and to raise public awareness about the asteroid impact hazard

A good time to talk about a project on my wish list. An orbital telescope devoted to finding  asteroids.

A wide field infrared scope much like WISE. But unlike WISE positioned at SEL1 or SEL2 so the earth isn't a major heat source.

A scope that can make simultaneous observations in visual wavelengths as well as infrared. This would tell us the asteroid's albedo from which we'd get a good estimate of size.

A scope that points towards the inner solar system. For various reasons asteroids within the earth's orbit are very hard to see from earth's surface. An orbital scope pointing towards the inner solar system would give us an inventory of a body of objects we presently know almost nothing about.

I was surprised and pleased to learn such a telescope had already been proposed. NEOCam. Principal Investigator Amy Mainzer.

We're presently getting a pretty good inventory of Chixculub size rocks. But Tunguska sized rocks are much harder to see. And there's a bunch more rocks this size. NEOCam would help us get a better handle on potential city killers.

Another potential NEOCam benefit: It could inventory potential asteroids for mining companies like Planetary Resources or Deep Space Industries.

NASA administrator Jim Bridenstine is enthusiastic about developing space as a source of resources and enabling economic growth.  He's show interest in lunar poles as low hanging fruit.

Near Earth Asteroids are also a low hanging fruit.  If Bridenstine's goal is to expand our economic activity into deep space, NEOCam is a great investment.

A few days ago Marshall Eubanks commented:

NEOCam is not in good shape. Over a year ago it was given one additional year of fairly minimal support. About all that was said about it in this month's SBAG meeting was that it "Continues in extended Phase A" - i.e., on life support.

I hope this changes. We need a telescope devoted to asteroid discovery.

Space Meow Boys

Sections of this long post:

1) Space Cadets
2) Space Meow Boys
3) Tom Murphy
4) James Nicoll
5) Charlie Stross
6) Opening A New Frontier Is Doable


Space Cadets


We are confined to a small, fragile planet. Being limited to a finite body of resources mean logistic growth. And we're rapidly approaching the ceiling to our logistic growth.

Opening a vast new frontier would allow growth for centuries or even thousands of years. Breaking free of Cradle Earth would be the most dramatic turning point in human history. If it’s possible then this goal is well worth pursuing.

But can we open the solar system to settlement and economic use? This is an open question in my opinion.

Some say space settlement is impractical. Be content with our limits, we’re told. Trying to push past our boundaries is a waste of time and we shouldn’t even try.

Civil, rational arguments are worth listening to. But some discussions are long on vitriol and short on math and physics.

Tarring With A Wide Brush

One dirty technique is tarring with a wide brush -- First find weak members in a group. Then hold up these members up as representative of the entire group. Give them a label.

Physics professor Tom Murphy does this. He holds up his clueless students as examples of space enthusiasts and tars us all with the label space cadets. Judging by the stories he tells, his students are some of the stupidest people on the planet. I suspect he teaches Astronomy 101 for Liberal Arts Majors.

Science fiction writer Charles Stross and book reviewer James Nicoll also like to use the label space cadet. They point to folks from Usenet who are long on wishful thinking and short on math skills. Their flavor of space cadet tends to be white and Libertarian.

Wrestling With A Pig

Friends tell me “Don’t wrestle with a pig. You both get dirty and the pig likes it.”  What they don’t realize is that I too am a pig. I love to wrestle in the mud!



I don't mind their dirty tricks. I'll do the same.

First I'll find nay sayers clueless in math and science. My label will be Space Meow Boys.


Space Meow Boys


Tom Murphy, James Nicoll and Charlie Stross are my examples of space meow boys.




Tom Murphy


Let's look at Murphy’s blog post Stranded Resources.

Murphy correctly puts a big emphasis on delta V and Tsiolkovsky’s rocket equation. But he sucks at calculating delta V. From his blog:

The next plot puts this in perspective, albeit only in simplified, approximate terms. The bottom of the plot represents the Earth’s ground. It takes 7.7 km/s of velocity to get to LEO (actually, it takes the equivalent of about 9.5 km/s because much effort is expended just climbing out, in addition to establishing the orbital speed). At 11.2 km/s, we’re free to take on the solar system.  The plot is based on minimum-energy Hohmann transfer orbits.



Each planet is represented by three dots: the top one being outside the planet’s grip in an identical solar orbit, the next one down at low-planet orbit (akin to LEO), and the lowest represents being at rest on the surface. For Saturn and Jupiter, these surface points are off the chart—so taxing is this requirement. And for these two, there’s no “there” there anyway to land on. Crudely speaking, we must have the means to accomplish all vertical traverses in order to make a trip. For instance, landing on Mars from Earth requires about 17 km/s of climb, followed by a controlled 5 km/s of deceleration for the descent. Thus it takes something like 20 km/s of capability to land on Mars, . . .

I bolded Murphy’s discussion of the Earth to Mars trip. Let’s look at his delta V.

He takes Earths 11.2 km/s escape velocity and adds in the ~6 km/s difference between Earth’s and Mars’  heliocentric orbits and then adds in Mars 5 km/s escape velocity. Which gives 22 km/s. Then Murphy leaves us with the impression he‘s being generous when he rounds down to 20 km/s

A first year aerospace student would cringe at Murphy’s bungled math. You don’t simply add Vescape and Vinfinity.

To get velocity of the hyperbolic orbit needed for TMI (Trans Mars Injection):

Vhyperbola = sqrt(Vescape2 + Vinfinity2)



A memory device is to think of Vescape and Vinfinity as the legs of a right triangle. Velocity of a hyperbolic orbit would be the hypotenuse.

Correctly patching conics get us 17 km/s from Earth surface to Mars surface

What About The Atmosphere?

Murphy points to a penalty imposed by Earth’s atmosphere:

It takes 7.7 km/s of velocity to get to LEO (actually, it takes the equivalent of about 9.5 km/s because much effort is expended just climbing out, in addition to establishing the orbital speed).

Yes, we suffer a loss of around 2 km/s to climb above the earth's atmosphere. There's some atmospheric friction as well as gravity loss during ascent. We'll give Murphy this 2 km/s. So our delta v budget goes up to 19 km/s.

But an atmosphere also offers the possibility of aerobraking. Is it possible Murphy hasn't heard of aerobraking? Or is he dishonestly focusing on the delta V penalties of an atmosphere while ignoring the benefits? The charitable judgement here is that Murphy is horribly clueless.

Aerobraking at the Mars end of an Earth to Mars trip can shave 6 km/s off the delta V budget. This takes our delta V budget down to 13 km/s. This is less than what it takes to park a satellite in geosynchronous orbit, something we routinely do.

Aerobraking at the Earth end of a Mars to Earth trip can shave 11 km/s off the delta V budget. This leaves a delta V budget of around 6 km/s for the Earth to Mars trip.

Grab That Asteroid!

Asteroid retrieval is a notion entertained by John S. Lewis, Planetary Resources, Deep Space Industries and others. If not retrieval of an entire asteroid, then retrieval of commodities from an asteroid.

Murphy argues against this using a ridiculous straw man scenario:

The asteroid belt is over 20 km/s away in terms of velocity impulse. If the goal is to use the raw materials for production on Earth or in Earth orbit, we have to supply about 10 km/s of impulse. We would probably try to get lucky and find a nickel-metal asteroid in an unusual orbit requiring substantially less energy to reel it in. So let’s say we can find something requiring only 5 km/s of delta-v. Our imagined prize will be a cube 1 km on a side, having a mass around 1013 kg. This is very small for an asteroid, but we need to moderate our ambitions. From a resource point of view, it’s still a lot. 
To get this asteroid moving at 5 km/s with conventional rocket fuel (or any “fuel” that involves spitting the mass elements/ions out at high speed) would require a mass of fuel approximately twice that of the asteroid. As an example, using methane and oxygen, (4 kg of O2 for every 1 kg of CH4), we would require two years’ of global natural gas production to be delivered to the asteroid (now multiply this by a large factor for the fuel to actually deliver it from Earth’s potential well). The point is that we would be crazy to elect to push the asteroid our way with conventional rockets.

Four things wrong this picture.

1) Murphy hasn't heard of NEAs? There are NEAs (Near Earth Asteroids) much closer to the Earth-Moon system. The Keck Report talks about NEAs that could be parked in a loose lunar orbit for as little as .17 km/s. 2006 RH120 was temporarily captured to the earth moon system with no delta V.

2) Murphy wants to use methane/oxygen bipropellant. This has an exhaust velocity of around 4 km/s in a vacuum. The Keck folks propose using xenon and Hall Thrusters. Exhaust velocity for this sort of ion engine can easily be 30 km/s.

3) A kilometer asteroid is far too large for practical rockets to retrieve. It would also be insanely dangerous. The Tunguska event likely came from an object between 60 and 200 meters in diameter. The Chixculub impact which wiped out the dinosaurs was thought to have been 10 to 15 kilometers. Perhaps a misdirected rock 1 kilometer in diameter wouldn't be an extinction level event. But it'd certainly cost trillions in property damage. The Keck folks talk about safety considerations at the bottom of page 15 of their report. They look at retrieving a 5 meter rock. Should a 5 meter rock fall earthward, it'd burn up harmlessly in the upper atmosphere.

4) Murphy assumes a metal rich asteroid. He could spend a few minutes Googling and find that water is the first commodity asteroid miners hope to exploit. Propellant not at the bottom of an 11.2 km/s gravity well would be a game changer that would reduce the cost of spaceflight. And cheaper spaceflight is a prerequisite to profitably exploiting asteroidal metals.

Plugging Murphy's 5 km/s delta V budget and 4 km/s exhaust velocity into the rocket equations tells use that we'd need more than two tonnes of propellant for every tonne of asteroid.

Plugging in .17 km/s delta V and 30 km/s exhaust velocity gives 6 kilograms of propellant needed to park a one tonne asteroid.



The fellow on the left is Tom Murphy. To the right is a self portrait.

Sometimes Murphy tries to excuse himself by pointing to his waffle words and furiously waving his hands. He seems to think words like "approximately" or "roughly" salvage his questionable claims.


Only 3 orders of magnitude off.

Refuel In Space?

The lunar cold traps are thought to have rich deposits of water ice as well as other volatile ices. These potential propellant sources are about 2.5 km/s from EML1 and EML2.

Here's some delta V maps focusing on EML1 and EML2:



There are also asteroid folks who hope to mine water from NEAs. See this Planetary Resources video or this Deep Space Industries video. Some NEAs are up to 40% water by mass and are only a small delta V nudge from being parked in lunar orbit. A water rich asteroid parked in lunar orbit would be even closer to EML1 and EML2.

What is Murphy's argument against refueling in space?

He tells us it'd take a lot of delta V to get propellant from Jupiter or Titan.

Since the large delta-v’s required to get around the solar system require a lot of fuel, and we have to work hard to lift all that fuel from the Earth’s surface, could we just grab hydrocarbons from Jupiter or Titan and be on our way? 
Let’s say you arrived in Jupiter orbit running on fumes, relying on the gassy giant to restock your coffers. In order to get close enough to Jupiter, you’ll be skimming the cloud-tops at a minimum of 42 km/s. Getting 1 kg of fuel on board will require you to accelerate the fuel to the speed of your spacecraft, at a kinetic energy cost of 885 MJ. The energy content of methane is 13 kcal/g, or 54 MJ/kg. Oops. Not even enough to pay for itself, energetically. Get used to Jupiter. And I have completely ignored the fact that you need marry two O2 molecules to each molecule of methane, meaning you actually get only 11 MJ per kilogram of total fuel. Utterly hopeless.

No shit, Sherlock. Knock yourself out beating up this straw man.


Tom Murphy's argument is perhaps the stupidest straw man ever.

Momentum Exchange Tethers

In the comments section for Stranded Resources, Monte Davis writes:

At the level of fundamental elegance, you can’t beat tethers: instead of throwing away momentum in exhaust, you just keep re-using it as payloads are slung around — assuming tethers at all sources/destinations and an abundance of payloads. Before that, make-up energy could be supplied by spinning up tethers slowly with a low-thrust solar-electric or nuclear-electric drive.

Murphy replies to Davis:

I don’t follow the first point about not throwing away momentum in the form of exhaust in a tether system. Without throwing away momentum, you can gain none (and go nowhere). If stranded on a frictionless lake on a sled piled with bricks, the only way off is to hurl bricks away. If the bricks are tethered to you, you may be able to move about as mass is redistributed, but the center of mass will be in the same place always.

Momentum isn't thrown away. It's exchanged.

An orbital tether would not sit motionless like a brick on a frozen lake. It would drop after catching a payload from a lower orbit. It would also drop when throwing a payload to a higher orbit.

However an orbital tether would rise after dropping a payload to a lower orbit. It would also rise when catching a payload from a higher orbit.

With two way traffic an orbital tether could balance momentum draining maneuvers with momentum boosting maneuvers and thus maintain an orbit without huge amounts of propellant.

Also as Davis mentions, a tether can use ion engines. Ion engines can easily have 30 km/s exhaust velocities while the best chemical is around 4.4 km/s. This is a much more efficient way to restore momentum. With low thrust engines it would take a long time to build momentum but that would be okay if there were weeks between tether maneuvers.

Monte Davis is a science writer and editor who's worked for Omni, Discover, Psychology Today and other publications. He's got a chemistry degree from Princeton. In space forums Davis usually plays the devil's advocate against would be space colonizers.

Murphy could have invested 4 or 5 minutes Googling momentum exchange tethers. But he blows off Monte Davis as if he's one of his clueless students in Astronomy 101 for Liberal Arts Majors.


James Nicoll


James Nicoll reviews science fiction. An old Heinlein chestnut is "If you can get your ship into orbit, you're halfway to anywhere." Nicoll attempts to play with this notion at More Words Deeper Hole.

Apparently the subject line I was going to use is offensive so I will go with "halfway to anywhere" 
james_nicoll
april 1st, 2012 
Suppose it's the future and further suppose that space tourism actually takes off enough that there are excursions to the Moon akin to what we see in Antarctica. Although probably not the 37,000 people a year you see headed to Antarctica because going to the Moon is going to a crapton more expensive. 
Further, suppose
it occurs to someone whose life centers on ferrying rich bastards back and forth to the Moon that the delta vee to go from Low Earth Orbit (LEO) to Low Lunar Orbit (LLO) is about 8 km/s. It's the same the other way, assuming no aerobraking at the Earth end (No aerobraking at the Earth end means big mass ratios or some kind of fuel depot in LLO). That's considerably more delta vee than it takes to to Mars from the Moon and it further occurs to them it might be fun on the next trip home to leave the tourists on the Moon and take an unsheduled excursion to Mars.


How would you go about adapting a vehicle designed to do the LEO-LLO trip to a LLO-Mars trip? 
The first big issue is going to be air. Assuming a dozen passengers and three crew, and about a week to the Moon and back, the ship probably doesn't have more than 105 person-days of O2. Fast but still reasonably delta-vee conservative orbit to Mars is about 180 days. 
I suppose, this being fiction, you could do it the other way: the would-be Marsnaut needs 180 person-days, therefore the LEO-LLO transfer ship carries a couple of dozen passengers and some crew. That will at least get the Marsnaut to Mars alive.

Delta V

Let's start with James' delta V budget.

it occurs to someone whose life centers on ferrying rich bastards back and forth to the Moon that the delta vee to go from Low Earth Orbit (LEO) to Low Lunar Orbit (LLO) is about 8 km/s.

According to the Wikipedia delta V chart James snagged, it's 4.1 km/s from LEO to L4/5 and then .7 km/s to lunar orbit.

4.1 + .7 = 4.8, not 8.



A direct route from LEO to LLO would be more like 4 km/s.

For hard SF folks, 8 km/s from LEO to LLO is a glaring error. But it's no biggie for the English Lit types that participate in James' forum. They don't even notice.

Aerobraking

James stipulates

It's the same the other way, assuming no aerobraking at the Earth end (No aerobraking at the Earth end means big mass ratios or some kind of fuel depot in LLO). That's considerably more delta vee than it takes to to Mars from the Moon and it further occurs to them it might be fun on the next trip home to leave the tourists on the Moon and take an unsheduled excursion to Mars.

Why on earth would James stipulate no aerobraking? This is a very standard technique. Is this because his premise rests on LLO to Mars taking less delta V than LLO to LEO?

Maybe he's heard Mars folks say LEO to Mars is less delta V than LEO to the moon. Which is true enough if aerobraking is used. With no aerobraking we'd need to do any where from .7 km/s for Mars capture to a 6 km/s burn for a soft landing. Or else we'd sail right past Mars back into a heliocentric orbit.

Hohmann Launch windows

Here's the biggest howler:

and it further occurs to them it might be fun on the next trip home to leave the tourists on the Moon and take an unsheduled excursion to Mars.

An unscheduled excursion?! Unless James’ ferry guys have a huge delta V budget, the ship's doing a Hohmann transfer. Windows for Earth to Mars Hohmann open once each 2.14 years. Lots of pre-planning is needed to take advantage of these rare windows. A trip to Mars isn't something you do at the drop of a hat.

As usually happens, James post stimulates a lively conversation. Most of the participants don't notice the howlers. The biggest concern seems to be sufficient air and food for the long trip.

A problem they seem oblivious to is radiation. An 8 month trip would expose the passengers to a lot more GCRs and solar flares than the 4 day LLO to LEO trip. Much more radiation protection would be needed. A few meters of water are often suggested to protect the passengers from GCRs. A few meters of water around the ship exterior would be a lot more massive than the air, food and drinking water James and his friends were obsessing over.

At one time I regarded James was one of the more numerate participants in science fiction forums. But he’s been spending too much time with SJWs and English Lit folks. Not that I dislike social justice or English literature. But if James wants to talk hard SF, he needs to revisit some of his math and physics textbooks.


Charlie Stross


Charlie Stross was one of the participants in the Nicoll post I just fisked. In that forum he goes by the handle autopope.  Nicoll’s lack of math and science savvy was not noticed by Stross or most of those commenting.

Stross was also crowing that physics professor Tom Murphy shared his opinions, as if that validates his views.

But we shouldn’t condemn Stross because of the company he keeps. Instead, let’s look at his High Frontier Redux.

It starts out noting the outer solar system and Alpha Centauri are far away and settling these regions isn’t practical. This is like saying the Americas were out of reach for the early humans in Africa. But the Americas became accessible after humans spread across Asia and reached the Bering Strait.

To show the Kuiper Belt is forever beyond reach, Stross needs to demonstrate intermediate destinations aren’t within reach.

Later he does argue against colonizing neighboring bodies. But starting off with the most difficult, furthest destinations is wasting the reader’s time.

Let’s look at Stross’ argument against developing the moon:

What about our own solar system? 
After contemplating the vastness of interstellar space, our own solar system looks almost comfortingly accessible at first. Exploring our own solar system is a no-brainer: we can do it, we are doing it, and interplanetary exploration is probably going to be seen as one of the great scientific undertakings of the late 20th and early 21st century, when the history books get written. 
But when we start examining the prospects for interplanetary colonization things turn gloomy again. 
Bluntly, we're not going to get there by rocket ship. 
Optimistic projects suggest that it should be possible, with the low cost rockets currently under development, to maintain a Lunar presence for a transportation cost of roughly $15,000 per kilogram. Some extreme projections suggest that if the cost can be cut to roughly triple the cost of fuel and oxidizer (meaning, the spacecraft concerned will be both largely reusable and very cheap) then we might even get as low as $165/kilogram to the lunar surface. At that price, sending a 100Kg astronaut to Moon Base One looks as if it ought to cost not much more than a first-class return air fare from the UK to New Zealand ... except that such a price estimate is hogwash. We primates have certain failure modes, and one of them that must not be underestimated is our tendency to irreversibly malfunction when exposed to climactic extremes of temperature, pressure, and partial pressure of oxygen. While the amount of oxygen, water, and food a human consumes per day doesn't sound all that serious — it probably totals roughly ten kilograms, if you economize and recycle the washing-up water — the amount of parasitic weight you need to keep the monkey from blowing out is measured in tons. A Russian Orlan-M space suit (which, some would say, is better than anything NASA has come up with over the years — take heed of the pre-breathe time requirements!) weighs 112 kilograms, which pretty much puts a floor on our infrastructure requirements. An actual habitat would need to mass a whole lot more. Even at $165/kilogram, that's going to add up to a very hefty excess baggage charge on that notional first class air fare to New Zealand — and I think the $165/kg figure is in any case highly unrealistic; even the authors of the article I cited thought $2000/kg was a bit more reasonable. 
Whichever way you cut it, sending a single tourist to the moon is going to cost not less than $50,000 — and a more realistic figure, for a mature reusable, cheap, rocket-based lunar transport cycle is more like $1M. And that's before you factor in the price of bringing them back ... 
The moon is about 1.3 light seconds away. If we want to go panning the (metaphorical) rivers for gold, we'd do better to send teleoperator-controlled robots; it's close enough that we can control them directly, and far enough away that the cost of transporting food and creature comforts for human explorers is astronomical. There probably are niches for human workers on a moon base, but only until our robot technologies are somewhat more mature than they are today; Mission Control would be a lot happier with a pair of hands and a high-def camera that doesn't talk back and doesn't need to go to the toilet or take naps.

In Situ Resources

Stross is right that human habitats in space would be massive. But he imagines every kilogram of a lunar habitat would be brought up from earth’s surface. Evidently Stross has never heard of in situ resources. At the lunar poles there are thought to be volatile ices — water ice as well as carbon dioxide ice and nitrogen compounds. Water and air to breathe could be extracted from local resources. Habs could be covered with regolith for radiation protection.

Stross acknowledges that robots could establish infrastructure on the lunar surface. And in fact this is what Spudis and Lavoie advocate.

In Situ Resources and Delta V

Besides building habs and infrastructure to extract life support consumables, robots could also build propellant mines. Stross didn’t bat an eye when Nicoll stated LEO to LLO is 8 km/s. It is likely this science fiction writer has no notion what role delta V plays in the rocket equation.

Mass propellant / mass payload = e(delta V/Vexhaust) - 1.

Exhaust velocity of hydrogen/oxygen bipropellant is about 4.4 km/s. Now 3/4.4 is very close to ln(2).

That means when using oxygen/hydrogen, every 3 km/s added to the delta V budget doubles over all mass.

Starting with 1 tonne rocket dry mass plus payload,
For 3 km/s you’d need 1 tonne propellant.
For 6 km/s you’d need 3 tonnes propellant.
For 9 km/s you’d need 7 tonnes propellant.
And so on.

Overall mass grows exponentially with increasing delta V. The legend of Paal Paysam illustrates the dramatic quantities exponential growth can give. Krishna challenged a king to a game of chess wagering a chess board with 1 grain of rice on the the first square, 2 grains on the second, 4 on the third and doubling each subsequent square. The king calculated the numbers for the first few squares and accepted. Here’s an illustration of Krishna’s wager:



Breaking the rocket equation’s exponent into chunks has a dramatic effect on the amount of propellant used. With each propellent depot, the delta V budget starts over:


We can start back to 1 grain of rice at each propellant depot.
Mount Everest is visible in this version, no longer covered with rice.


Delta V from earth’s surface to LEO is about 9.5 km/s. LEO to lunar surface is about 6 km/s. The additional 6 km/s boosts four fold the mass that needs to be parked in LEO.

If the ship could refuel in LEO, that would cut GLOW (Gross Lift Off Weight) four fold.

Here’s a delta V map focusing on EML2 and LEO. Moon to LEO is about 3 km/s using aerobraking.




But savings on propellent isn’t the chief advantage here. With an extraterrestrial propellant source, inter orbital tankers and ferries could move between orbits without ever suffering the extreme conditions of an 8 km/s re-entry into earth’s atmosphere.

Also with delta V budgets on the order of 4 km/s, inter orbital vehicles can devote a higher mass fraction to structure. Present day upper stages have less mass fraction than an aluminum Coke can. Which makes durable structure and adequate thermal protection very difficult if not impossible.


A racing bike vs a mountain bike.
With a racing bike we want to minimize mass.
But a racing bike is fragile while a mountain bike is durable and rugged.
When an upper stage has a 4% dry mass fraction, durability is not an option.

Elon Musk and Jeff Bezos seem well on their way to developing economical, reusable booster stages. Bezos wants to help establish lunar propellant mines. If Bezos, Bridenstine et al successfuly export lunar propellant to LEO, upper stages could refuel before re-entry into the atmosphere. Reuse of upper stages is much more plausible if re-entry velocity is 4 km/s or less.

Space Elevators

Stross mentions the possibility of Space Elevators.

Arthur C. Clarke popularized the notion with his novel Fountains of Paradise. Clarke, Asimov and Heinlein were writers from the great generation. They had some physics and tech savvy as well as an optimistic can-do attitude.

Baby boomer SF writers are more about bleak dystopias and cautionary tales. Like main stream pop culture they rely on sex and glorifying substance abuse to sell their product. With a few exceptions, SF writers from my generation tend to suck at math and physics. Hopefully younger science fiction writers will pick up the mantles of Heinlein and Clarke.

A space elevator was a good idea in the time of Clarke. Since then we’ve massive amounts of junk into Low Earth Orbit (LEO). Here is a panel from the Hubble telescope that spent 14 years in LEO:



See this Space Stack Exchange discussion on orbital debris.

The extreme height of a space elevator gives it enormous cross sectional area. Much more cross section than the panel pictured above. So even if we could manufacture long strands of Bucky tubes with insanely high tensile strength, the elevator would be severed by impacts.

However full blown Clarke towers have smaller cousins: orbital tethers. Being a lazy baby boomer writer, Stross seems content to rehash tired 1970s SF ideas. It is possible Stross has never heard of orbital tethers.

Orbital tethers can be placed in orbits relatively free of debris. They would be much shorter than a full blown Clarke Tower and would suffer much less stress. They could be made from existing materials like Zylon. I talk about orbital tethers at Trans Cislunar Railroad. Given two way traffic, a tether could harvest up momentum from higher orbits and trade it with the down momentum of lower orbits. Thus with two way traffic a tether could impart delta V with little expenditure of energy and propellant.

Orbital tethers could also be anchored on Phobos and Deimos.

Given tethers of modest mass, payloads can be exchanged between Phobos and Deimos via a Zero Relative Velocity Transfer Orbit (ZRVTO).



Given a somewhat more substantial tether, a Phobos tether could throw payloads down to a 1 A.U. perihelion (in other words, a transfer orbit to earth) or to a 3 A.U. aphelion (in other words a transfer orbit to the Main Belt).

An upper Phobos tether capable of launching payloads to various regions of our solar system needn't be that massive.



A Phobos tether extending to Mars upper atmosphere would drop payloads into Mars atmosphere at .6 km/s. About mach two, the Concorde Jet would routinely do this through a much thicker atmosphere. This about 1/10 the velocity landers from earth normally enter Mars' atmosphere. A Phobos tether descending to Mars' upper atmosphere isn’t practical using Zylon but would certainly be doable if they manage to manufacture long lengths of Bucky tubes.

Summary of Stross' Errors

Stross gives us numbers assuming all propellant and hab mass comes from earth's surface.

Using in situ resources most of the hab mass can be made from materials at hand.

More importantly there's the possibility of in situ propellant. This can drastically cut delta V budgets. Which cuts propellant and energy needed. It also makes robust, reusable vehicles possible.

Momentum exchange tethers are doable. This would further reduce energy and propellant needed to travel between space destinations.


Opening A New Frontier Is Doable


It is possible to establish infrastructure that would greatly reduce the cost of traveling about in space.

Yes, it would be expensive but it is doable. Dennis Wingo's book Moonrush documents several examples of government/private enterprise partnerships establishing massive transportation and communication infrastructure. The trans continental railroad was such a collaboration.

NASA administrator Jim Bridenstine has expressed his desire to work with SpaceX and Blue Origin to establish lunar and cislunar infrastructure. It is possible this could come to pass.

But the effort would have better prospects for adequate funding if the public perceived it as possible. The space meow boys have used bad math and silly straw man arguments to strengthen the public perception that this is pie in the sky.

The first steps towards opening the space frontier would be establishing infrastructure on other bodies. Semi-autonomous tele-robots are dropping in price while becoming more capable. British Petroleum has used R.O.V.s to build oil wells on the sea floor. It is possible to build the initial space infrastructure without a human presence.

Once robots have established infrastructure to extract propellant and keep humans alive, the cost of human presence plummets.

Why does Murphy argue so vehemently against a new frontier? He's worried that we'd be okay with trashing the earth if we had the option to move.  Bill Maher makes the same argument.

Maher and Murphy are giving us a false dichotomy. We can do both. We need to work to preserve our home as well as open new frontiers. Space advocates are more aware than the average person that our precious planet is finite and fragile.

For example Musk is also working on solar energy and electric cars in addition to his rockets. Bezos is advocating moving destructive mining and manufacturing out of our ecosphere.

Musk and Bezos are doing more for a sustainable future than a million space meow boys.


Wednesday, April 25, 2018

Tahoe recreational math conference

I've been invited to talk at a recreational math conference at Lake Tahoe April 28.

I will talk about the family of conics associated with a space elevator or a vertical orbital tether.

The presentation combines stuff from a number of my earlier blog posts. Here is the pdf I will use for this talk.

A screenshot from the cover of the booklet.

Pages 2 — 5 -- includes the visualizations I use to remember expressions for gravity and centrifugal acceleration. Which I use to show how canonical units are a whole lot easier to use than kilograms, kilometers and so on. For example the orbital periods or an orbit whose semi-major axis is k A.U. will have period k3/2 years. An asteroid whose semi-major axis is 4 A.U. will have a period of 8 years. A 9 A.U. semi-major axis gives a period of 27 years.

Pages 13 — 14 -- I believe my general method  for finding a ZRVTO orbit between tethers is new and original. If anyone knows of this method appearing in earlier publications, please give me a heads up.

Page 14 -- Most moons we know of are in nearly circular tide locked orbits. For such moons beanstalks through Planet-Moon L1 and L2 are plausible. I believe this would be a wonderful science device. But today's science fiction writers seems oblivious to any of the Lagrange points other than L4 and L5.

Pages 15 and 16 -- Shameless self promotion. I don't get any royalties on the Dover coloring books, I was paid a straight fee of $100 per page. Regardless, I'd like to see the books sell. A Dover editor placed his bets on me and I hope he is rewarded for this gamble. I do make money on T-shirt sales though. My T-shirts are available at Ajo-Copper-News.com  .



Friday, June 9, 2017

Zylon Mars Elevator

Mars Elevator With Conventional Materials

Mars spins nearly the same rate as earth (about a 24.62 hour day). Mars has about 1/9 earth's mass. At 17,000 kilometers, altitude of Mars synchronous orbit is less than half the altitude of geosynchronous orbit (about 36,000 kilometers).

These considerations have led some Mars enthusiasts to claim a Mars elevator made of conventional materials is possible. No bucky tubes or other science fiction material is needed, Kevlar will do. Is this true? I will take a look using Chris Wolfe's spreadsheet.

Safety Factor

In earlier blog posts using Wolfe's spreadsheet I used a safety factor of 1, a razor thin margin. The slightest scrape or nick will make the tether break. This is like drawing a pentagram to summon the demon Murphy's Law. No sensible entity would risk expensive payloads on such a narrow margin. Much less human lives. I hope to revise my earlier blog posts to include more sensible safety margins.

In later blog posts I looked at scenarios using a safety factor of 3. With this margin a portion of tether can lose up to 2/3 of it's mass without breaking.

In this post I'll use tables looking at a range of safety factors.  With a safety factor of 2, I cut tensile strength in half. A safety factor of 3 cuts tensile strength to a third. Which is a lot like cutting exhaust velocity in the rocket equation. Increasing an exponent can make tether thickness sky rocket.

Mars Equator to Mars Synchronous Orbit

This is the lower part of a Mars elevator. It exerts downward newtons that need to be balanced with upward newtons from elevator mass above Mars synchronous orbit.

Safety
 Factor 
Zylon
Taper
Ratio
Tether to
Payload
 Mass Ratio 
1
13
154
2
162
3191
3
2016
51824

Payload is mass of elevator car as well as elevator car's contents. The elevator car will need to include motors and power source.

Mars Synchronous to Sub Deimos Elevator Top

Elevator top is set 50 kilometers below Deimos' periapsis. This is to avoid collision. The counterweight and tether above Mars synchronous orbit must counterbalance the downward force of the lower elevator.

Safety
 Factor 
Zylon
Taper
Ratio
Tether to
Payload
 Mass Ratio 
Counterweight
to Payload
 Mass Ratio 
1
1.02
38
1200
2
1.03
955
14800
3
1.05
1761
180000


The Whole Shebang

Safety Factor 1

Assuming lifting a 10 tonne elevator car and contents from Mars' surface and given a safety factor of 1, we'd need 10 * (38 + 154) tonnes of tether material. That'd be 1,920 tonnes of Zylon. Perhaps worthwhile if the elevator had a vigorous through put. I think these are the numbers Mars enthusiasts are talking about when they talk about Mars beanstalks made of Kevlar.

Also needed would be a 12,000 tonne counterweight. That's about thirty times the mass of the I.S.S.. This to lift a 10 tonne elevator car from Mars' surface? The need for a stud hoss counterweight sinks the argument for a Mars elevator, in my opinion.

Safety Factor 2

10 * (162 + 955) = 11170. About 11 thousand tonnes of Zylon to lift a 10 tonne elevator car and contents.

We'd need a nearly 150,000 tonne counterweight.

I think it's pretty obvious a Zylon Mars elevator with a safety factor of two isn't worthwhile. I'm not going to bother looking at a safety factor of 3.

Benefits

The elevator top is moving at about 1.7 km/s. It needs another 1.6 km/s to achieve Trans Earth Insertion (TEI). From the surface of Mars it takes about 6 km/s for TEI. So the elevator cuts saves about 4.4 km/s off of trips to earth.

Obstacles

Given a sensible safety factor, a Zylon tether would need to be much more massive than the payload. The counterweight mass would dwarf the payload mass.

Mars neighbors the main asteroid belt. Some rocks from the belt make their way to Mars neighborhood. Collision with asteroidal debris could cut the tether. Given this elevator's 20,000 km length and healthy taper ratio, there is a large cross sectional area. This increases likelihood of an impact.

Also there is a chunk of Debris named Phobos which crosses the elevator's path every 10 hours or so.

Comparison to Phobos Elevator


A Phobos elevator dropping to Mars' upper atmosphere and extending to Trans Ceres insertion is about 13,700 km. This about 6,000 km shorter than the Mars elevator described above. It also has a smaller taper ratio. This makes for a smaller cross sectional area to intercept debris. Being anchored at Phobos, this elevator won't collide with Phobos. The top is well below Deimos. orbit.

This tether can provide Trans Ceres Insertion as well as Trans Earth Insertion.

It takes about a .6 km/s suborbital hop for a Mars ascent vehicle to rendezvous with this tether foot.

Using a safety factor of 1, the upper Phobos tether has a 3.21 payload to mass ratio. The lower Phobos tether has a tether to payload mass ratio of about 16.1. So from top to bottom, about twenty times the payload mass is needed in Zylon.

The Phobos takes about 1/10 of the Zylon mass for a mars elevator with a safety factor of one.

A sub Deimos Mars elevator can't throw payloads above Mars escape velocity.
But with higher taper ratio, it'd take ten times as much zylon mass than a Phobos elevator.
This is with a safety factor of 1.
A Zylon Mars elevator with better safety factors is impractical.

I hope to revisit the upper Phobos tether and lower Phobos tether pages and include safety factors of 2 and 3. I suspect with a higher safety factor that a Zylon tether from Phobos to Mars upper atmosphere may not be feasible.










Wednesday, February 1, 2017

Matrices

Vectors

Rotation Matrix
Proportional Scaling Matrix
Non Proportional Scaling Matrix
Shear Matrix
Reflect Matrix

Determinant of a Matrix

Lorentz Transform Matrix


Vectors


Vectors are a way to describe point locations with numbers. Vectors can be used to build simple shapes like a cube or just about any shape you can imagine.

We do lots of stuff to these vectors with matrix multiplication.  We can grow, shrink, spin, stretch, squeeze, tilt and flip these guys.



First we'll look at the things you can do to vectors on a plane with computer drawing programs like Adobe Illustrator. Below are some items from the Illustrator tool box that employ matrices.



Rotation Matrix



Rotating a polygon doesn't change it's area. The area remains the same. The determinant of this matrix is 1.



Proportional Scaling Matrix



Doubling size as well as height boosts a polygon's area by a factor of four. This determinant of this matrix is 4.



Non Proportional Scaling Matrix



This matrix stretches the width to twice the original and squeezes the height to half of what it was. Overall the area is unchanged. The determinant of this matrix is 1. However the determinant of a non proportional scaling matrix can be more or less than 1.



Shear or Skew Matrix


When I was using Macromedia Freehand, the graphics program called this transformation "skew". Then Adobe ate Macromedia and I was forced to use Adobe Illustrator. Illustrator calls it "shear".


This transformation transforms a horizontally aligned rectangle to a parallelogram with same base and height. Area remains unchanged. The determinant of this matrix is 1.


Flip Matrix



Making the first term in the main diagonal negative flips polygons about the y axis. Making the lower right term negative would flip polygons about the x axis.

Determinant is -1. Not sure what that means geometrically but absolute value of the area remains the same.

Illustrator Tool Box



Determinant of a Matrix


Below is a general 3x3 matrix multiplied by each of the usual basis vectors in 3-space.
Notice the first basis vector is transformed into the first column of the matrix, the 2nd basis vector is transformed into the second column, and so on.

The 3 basis vectors form edges of a unit cube. Multiplying each of the vertices of this unit cube by our general matrix, we get a parallelepiped with edges (a, b, c), (l, m, n) and (x, y, z).

Of course the volume of the unit cube is one cubic unit. To find the volume of the transformed parallelepiped we take the determinant of the matrix

There are some matrices that don't change the size or shape of the objects they transform. Rotation matrices, for example. These have a determinant of 1. Matrices that don't change the size but flip the chirality of an object (say, change a left shoe into a right shoe), have a determinant of -1.

Here is a nice vid on the determinant of a matrix.




Lorentz Transformation


In ordinary Euclidean space, a point (x, y, x)'s distance from the origin would be
sqrt(x2 + y2 + z2 ), a metric easily arrived at with the Pythagorean theorem.

But the time space manifold we dwell in is a little strange. The metric is
sqrt(-t2 + x2 + y2 + z2 ). One of these dimensions is not like the other one.

In ordinary Euclidean space, changing Point Of View (POV) entails a translation and/or a rotation. In our space time, changing POV entails a Lorentz Transformation.

Adam Zalcman did a nice job of portraying the Lorentz transformation as a matrix. Here is a screen capture from his physics stack exchange answer:


A Lorentz matrix for a 2 dimensional Minkowski space looks like this:


Above is our two dimensional Minkowski space. As they move through time inhabitants can move either right or left. The ship leaves earth in the present. A year later it has moved half a light year to the right. It is moving .5 c.

Pluggin .5 c into our Lorentz transform matrix we get:

Transforming our Minkowski space with this matrix we get:
The ship's world line has been shoved to the left. From the ship passengers' point of view, they aren't moving. Also they perceive .866 years have elapsed, not a full year. The earth's world line has also been shoved to the left. From the ship's POV the earth is moving .5 c to the left.

Note that the diagonals remain in the same place. Earth folks as well as ship passengers both perceive light photos to be traveling at 1 c (c is the speed of light).

While the transformation stretches along one diagonal, it also squeezes along another diagonal. So the area remains the same. Determinant of this matrix is one.

When I first saw the transformed coordinate system I was thinking "Wait a minute. Earth is now more than half a light year away and only .866 years have passed on the ship. Seems like earth is going more than .5 c.  My mistake was in using the word "now". What were simultaneous events from one frame are no longer simultaneous.


Note that from the ship's P.O.V. Earth's clock is running slower. This is possible because simultaneous events along worldlines change depending on which frame the viewer's in.


Here is an animation showing different transforms where the ship's speed varies from -.9 c to .9 c.


Winchell Chung of Atomic Rockets describes a scene from a Heinlein novel where a student asks:

     “Mr. Ortega, admitting that you can’t pass the speed of light, what would happen if the Star Rover got up close to the speed of light—and then the Captain suddenly stepped the drive up to about six g and held it there?”
     “Why, it would—No, let’s put it this way—” He broke off and grinned; it made him look real young. “See here, kid, don’t ask me questions like that. I’m an engineer with hairy ears, not a mathematical physicist.” He looked thoughtful and added, “Truthfully, I don’t know what would happen, but I would sure give a pretty to find out. Maybe we would find out what the square root of minus one looks like—from the inside.”

Let's take a look at world lines where one rocket is moving .5 c to the left and the other is moving .5 c to the right. At first glance it'd seem like the rocket moving to the left would be moving the speed of light with regard to the other rocket.


Transform the scene on the left to the orange ship's point of view. From the orange ship's P. O. V., the purple ship is moving .8 c to the left. The Lorentz transformation doesn't shift the purple ship all the way to the edge of the light cone. (Click on image to get a larger version).


From the point of view of each world line, his immediate neighbors are moving either .5 c to the left or .5 c to the right. The arrowhead on each line corresponds to the passage of one year from that world line's point of view. The horizontal line indicates simultaneous events from the P.O.V of the central world line after one year. These trace out a hyperbola with the edges of the light cone as asymptotes.

If the above lines were a cone, plane of simultaneous events would cut the cone along a circle and the world lines would pierce that circle in points closer and closer the edge as the world lines approached c. This would be a Poincare disk.

M. C. Escher's Circle Limit prints are based on Poincare disks.


Each angel or demon on this disk perceives themselves to be at the center while their neighbors shrinking towards the boundary of this world as they grow more distant. So it would be with Mr. Ortega's student who would step on the gas when he's moving .999 c. He'd just shift his position to the another part of the disk and he would be no closer to the edge.