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
where
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.

Alternatives?


Above is a Johnny Robinson cartoon from the National Space Society's publication.

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.



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 earns Neil an F for fail. 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.



Here Tyson seems to be talking about the so called Interplanetary Transport network. I went over this in Potholes on the Interplanetary Super Highway. The Weak Stability Boundaries (WSBs) emanating from Sun Earth Lagrange 1 (SEL1) or Sun Earth Lagrange 2 (SEL2) won't get you very far from earth's orbit. Same goes for the Sun Mars L1 and L2. Mass parameters between the sun and the tiny rocky planets are too small to be of much use.

The lack of an Oberth benefit largely compromises the advantage of SEL2's high location. A chemical Trans Mars Insertion (TMI) from Low Earth Orbit (LEO) is about 3.6 km/s. From SEL2, a direct burn to TMI takes about 3 km/s. An advantage of .6 km/s? No. When you consider that it takes a few months plus a 3.1 km/s LEO burn to reach SEL2, there is no advantage.

From SEL2 it's possible to do a multi burn TMI that enjoys the Oberth benefit. As described in What About Mr. Oberth?, a small burn can drop a craft deep in earth's gravity well where a perigee burn would confer a large Oberth benefit. But it takes a few months to fall earthward from SEL2. About the same Oberth benefit could be enjoyed by dropping from EML2 and this only takes 9 days.

Unlike SEL2, EML2 is close to the lunar cold traps or, potentially, asteroids parked in lunar orbit. These are possible sources of life support consumables and propellent. Close to EML2 in terms of time and distance as well as delta V.

I wish Tyson had recalled this article when he made his trailer for The Martian. The L1 and L2 regions sit on the edge of a planetary gravity well. Going from SEL2 to SML1 cuts climbing in and out of two deep wells. Weir's 124 day journey would have been plausible. But the Hermes would still have had a substantial delta V budget. It's not possible to ride the currents of space to "drift" from SEL2 to SML1.

Tyson on "idiot doctors"


Tyson says:

Somebody's diagnosed with terminal cancer. The doctor says you got 6 months to live. … Go to a 2nd doctor, you got 5 months to live. Go to a 3rd doctor, 7 months to live.  … What happens? You're alive a year later. … 3 years later the cancer's in remission. 5 years later, it's gone from your body.

You happen to have been a religious person … Here's what's astonishing - if you are that person you are more likely to believe that God cured you … than that you had 3 idiot doctors diagnose you.
Does someone surviving stage 4 cancer demonstrate the idiocy of doctors? No.

A competent doctor gives his patient the odds. He tells his patient the survival rates of other people in his condition.

Avi Bitterman notes:

When a doctor gives you a diagnosis that you have cancer and you are going to die in X amount of time, what the doctor is doing is giving you an average time of death with an amount of certainty given by the 95th or 99th percent confidence intervals of the time of death since diagnosis for people who were in your same situation. As with any bell-curve distribution there's bound to be outliers. That's just how a bell-curve works. If you manage to outlive that, it doesn't mean anything magical happened, but contrary to what Tyson suggests, it doesn't mean the doctor who made this prediction is stupid either, it just means a bell-curve works the way a bell-curve works. One would think this is something Tyson would know a thing or two about.
Indeed, statistics and bell curves crop up in most fields of science.

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.

Are there more Tyson science 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 science bloopers, drop me a line.


Thursday, January 7, 2016

Deimos Tether

This is a fourth in a series of blog posts looking at various tethers using Chris Wolfe's model.

50 kilometer Deimos tether - minimum length to remain aloft.

Mars-Deimos L1 and L2 are about 14 kilometers from Deimos' surface. Another 26.5 kilometer length extended past these points would balance. Extending the tether 50 kilometers either way along with a counterweight would provide enough tension for the elevators to stay aloft.

Zylon taper ratio is 1. Tether to payload mass ratio: about .01. A ten kilogram tether could accommodate a thousand kilogram payload.

Benefits

There is no net acceleration at L1 and L2, so docking at ports at these locations would be like docking with the I.S.S.

This first step could serve as a scaffolding additional tether infrastructure could be added onto.

2942 kilometer lower Deimos tether - transfer to Phobos tether

Given an ~1000 upper Phobos tether and a ~3000 lower Deimos tether, it is possible to move payloads between the two moons with almost no reaction mass. The tether points connected by the ellipse match the transfer ellipse's velocities. See my Upper Phobos Tether post.



Zylon taper ratio: 1.01. Tether to payload mass ratio: .04. A one tonne tether could accommodate a twenty-five tonne payload.

Benefits

The idea of ion driven interplanetary vehicles excite me. The Dawn probe has demonstrated ion rockets are long lived and amenable to re-use. An ion rocket's fantastic ISP means a lot more mass fraction can be devoted to payload.

However ion rockets have pathetic thrust. They suck at climbing in and out of planetary gravity wells.

Here Mark Adler talks about ion rocket trajectories:

The fictitious Hermes from Andy Weir's The Martian can do 2 mm/sec^2 acceleration. Due to the need for a high alpha, I regard the Hermes as medium implausible but I will go with that number.

At Deimos' distance from Mars, gravitational acceleration is about  80 mm/s^2. The Hermes' acceleration over Mars gravitational acceleration at that orbit is about 1/40. A small fraction but a lot larger than the 10^-3 fraction Adler mentions.

Deimos moves about 1.35 km/s about Mars. With an impulsive chemical burn, it would take about .56 km/s to achieve escape. But with a 2 mm/s^2 acceleration, it would take about 5 days and and .8 km/s to achieve escape.

To spiral down to low Mars Orbit, it'd take Hermes more than 17 days and 3 km/s. So the Deimos rendezvous says about two weeks and more than 2 km/s delta V.

Once in heliocentric orbit, it is the sun's gravitational acceleration that we put in the denominator. Here is a chart of gravitational acceleration at various distances from the sun:


If the rocket's acceleration is a significant fraction of central body's acceleration, we can model burns as impulsive. The trajectory would be more like an ellipse than a spiral. At earth's distance from the sun., Hermes 2 mm/s^2 acceleration would be about a third the sun's gravity. At Mars, it's about four fifths. In the asteroid belt, Hermes acceleration exceeds acceleration from sun's gravity.

Ion rockets may not be great for climbing in and out of planetary gravity wells. But they're fine for changing heliocentric orbits, especially in the asteroid belt and beyond.

Saturday, January 2, 2016

Upper Phobos Tether

This is third in a series of posts that rely on Wolfe's model of tethers from tide locked moons. As with the Lower Phobos Tether post, I will look at possible stages of this tether examining tether to payload mass as well as benefits each stage confers.

7 kilometer upper Phobos tether - tether doesn't collapse but remains extended

I used Wolfe's spreadsheet to find location of tether top where tether length Phobos side of L2 balances the length extending beyond L2. This occurs 6.6 kilometers from the tether anchor. Having the tether extend 7 kilometers is sufficient to maintain tension.


Docking with a facility at the L1 or L2 regions is easier than landing on Phobos. In the words of Paul451: "Instead of a tricky rocket landing at miniscule gravity on a loosely consolidated dusty surface, you just dock with the L1-hub of the ribbon (same as docking with ISS), transfer the payload to the elevator car and gently lower it to the surface. Reverse trip to bring fuel from Phobos to your ship (Assuming ISRU fuel is available on Phobos.)"

937 kilometer upper Phobos tether - transfer to Deimos tether

Given tethers from two coplanar moons tidelocked to the same central body, it is possible to travel between the two moons using nearly zero reaction mass.

Above I attempt to show how peri-aerion and apo-aerion of elliptical transfer orbit matches velocity of the tether points this ellipse connects. Tether Vs are red, transfer ellipse'sVs are blue.


Above I try to explain the math for finding the tether lengths from Deimos and Phobos.

Trip time between the two tethers is about 8 hours.

Zylon taper ratio for a 937 kilometer tether length is 1.02. Tether to payload mass ratio is .0448. Or the tether is about 1/20 the mass of the payload.

I'll look at the Deimos tether in a later post.

Benefits

Easy travel between Deimos and Phobos is a benefit in itself. 

But this would be a huge help to ion driven Mars Transfer Vehicles.

I like the notion of reusable ion driven MTVs. Ion engines have have great ISP thus allowing a more substantial payload mass ratio. However they have pathetic thrust. Andy Weir's fictional Hermes spacecraft can accelerate at 2 millimeters/sec^2. Which actually is very robust ion thrust. However ithis is only medium implausible. Low thrust means little or no planetary Oberth benefit. Plus a lo-o-o-ng time to climb in and out of planetary gravity wells.



300 km above Mars surface in low Mars orbit, gravitational acceleration is about 3 meters/sec^2. For a 300 km altitude low earth orbit, gravitational acceleration is about 9 meters/sec^2. 2 mm/s^2 acceleration is less than 10^-3 of the gravitational acceleration at initial orbit velocity in both these case. However I will be kind and go with Adler's .856 * initial orbit velocity.

At 2 millimeters/s^2 it would take Hermes 38 days to spiral out of earth's gravity well from low earth orbit and 17 days to spiral out of Mars gravity well. Most of the slow spiral out of earth's gravity would be through the intense radiation of the Van Allen belts.

I was very disappointed when Neil deGrasse Tyson's trailer had Hermes departing from low earth orbit and arriving in Mars' orbit 124 days later.

Besides adding 10 km/s to the delta V budget, climbing in and out of gravity wells would add about two months to Hermes' trip time. Tyson's video describes an impossible trajectory.  I wish he'd fact check himself with the same enthusiasm he applies to others.

It would be much better for Hermes to travel between the edges of each gravity well. At least as close as practical to the edge. In earth's neighborhood, Hermes could park at EML2 between trips. In Mars' neighborhood, parking at Deimos would save a lot of time and delta V. From Deimos, astronauts and payloads can transfer to Phobos and then to Mars surface. In this scenario, Hermes' 124 day trip from earth to Mars is plausible.

2345 kilometer upper Phobos tether - Mars escape

If anchor in a circular orbit, escape velocity can be achieved if tether top is at a distance 2^(1/3) anchor's orbital radius. I try to demonstrate that here. Phobos is in a nearly circular orbit. To achieve escape, the tether would need to be 2435 kilometers long.

Zylon taper ratio: 1.11. Tether to payload mass ratio: .23. A little more than 1/5 of the payload mass.

Benefits:

Achieve mars escape.

6155 km kilometer upper Phobos tether - To a 1 A.U. heliocentric orbit

A tether this long can fling payloads to a 1 A.U. heliocentric orbit, in other words an earth transfer orbit.

Taper ratio: 1.8. Tether to payload mass ratio 1.6. The tether mass is nearly double payload mass.

Benefits

Catch/throw payload to/from earth.

7980 kilometer upper Phobos tether - to a 2.77 A.U. heliocentric orbit.

Zylon aper ratio: 2.53. Tether to payload mass ratio 3.21. A little more than triple payload mass.

Benefits:

2.77 A.U. is the semi major axis of Ceres. A tether this long could catch/throw payload to/from Ceres. But this doesn't take into account plane change because of Ceres inclination.

Even with plane change expense, this tether could be very helpful for traveling to and from The Main Belt.








Thursday, December 24, 2015

Lower Phobos Tether

A Phobos tether can be built in increments, it is useful in the early stages. So there's no pressing need to build a huge structure overnight. I will look at various stages of a Phobos tether, examining mass requirements and benefits each length confers. To model the tether I am using Wolfe's spreadsheet. I will use Zylon with a tensile strength at 5,800 megapascals and density of 1560 kilograms per cubic meter. Here is the version of the spreadsheet with Phobos data entered.

7 kilometer lower Phobos tether - tether doesn't collapse but remains extended

At a minimum, the lower Phobos tether must extend far enough past Mars-Phobos L1 that the Mars-ward newtons exceed the Phobos-ward newtons. This will maintain tension and keep the elevator from falling back to Phobos.

I used Wolfe's spreadsheet to find location of tether foot where tether length Mars side of L1 balances tether length from Phobos to L1. That occurs when tether foot is about 6.6 kilometers from tether anchor:


So going past that a ways will give a net Marsward force.


At this stage tether to payload mass ratio is about .01. The tether length exerts negligible newtons compared to payload force. Therefore a payload descending the tether to Phobos' surface would exert enough force to collapse the tether, especially as it nears Phobos' surface. So a counterbalancing mass would be needed at the tether foot.

Benefits

Escape velocity of Phobos is about 11 meters/sec or about 25 miles per hour. A small rocket burn would be needed for a soft landing. This burn could kick up dust and grains of sand, some of which could achieve orbit. This would create an annoying debris cloud.

However a spacecraft could dock with a station at Mars Phobos L1 much the same way we dock with the I.S.S.  Payloads could then descend the tether and arrive at Phobos without kicking up debris.

It would also allow low thrust ion engines to rendezvous with Phobos.

It would also serve as a foundation which can be added to.

It would take a Mars Ascent Vehicle about 5 km/s to leave mars and rendezvous with this tether. Trip time would be about two hours, so the MAV could be small.

From this Phobos tether, a .55 km/s burn can send drop a lander to an atmosphere grazing periapsis. Aerobraking can circularize to a low Mars orbit moving about 3.4 km/s. If Phobos is capable of providing propellent, much of that 3.4 km/s could be shed with reaction mass.

In contrast, a lander coming from earth will enter Mars atmosphere at about 6 km/s. Since it takes about 14 km/s to reach this point, the lander will not have reaction mass to shed the 6 km/s. For more massive payloads like habs or power plants, shedding 6 km/s in Mars atmosphere is a difficult Entry Descent Landing (EDL) problem.

87 kilometer lower Phobos tether - copper pulls it's own weight

It would be nice to have power to the elevator cars. However copper only has a tensile strength of 7e7 pascals and density of 8920 kilograms per cubic meter. Have copper wire along the length of the Zylon tether would boost taper ratio. Using the spreadsheet, I set tensile strength and density to that of copper and lowered the tether foot until I got a taper ratio of 1.1. That gives a length of about 87 kilometers.


Benefits

Along this length of the tether, copper pulls it's own weight, as well as supports the payload. A massive power source can be placed at L1 -- at L1 there are no newtons either Phobos-ward or Mars-ward. A copper only tether of this length would be about .2 times that of payload mass.

Elevator cars can ascend this length without having to carry their own solar panels and battery.

If descending from L1 Mars-ward, Mars' gravity can provide the acceleration and no power source is needed.

Of course copper wires can be extended further but this would boost taper ratio as well as tether mass to payload mass ratio.

From this tether foot, it takes .54 km/s to drop to an atmosphere grazing orbit. Trip time is about two hours.

1,400 kilometer lower Phobos tether - release to an atmosphere grazing orbit


With Zylon, tether to payload mass ratio is .11. The tether mass is still a small fraction of payload mass.

Benefits

Releasing from the foot of this tether will send a payload to within a 100 kilometers of Mars' surface. Skimming through Mars upper atmosphere each periapsis will shed velocity and lower apoapsis.

Low Mars orbit velocity is about 3.5 km/s. The payload arrives at 4.1 km/s.

4,300 kilometer lower Phobos tether - payload enters atmosphere at 3 km/s.

With Zylon, tether to payload mass ratio is 2.55. Tether mass is almost triple payload mass.

Benefits

At 4,300 kilometers from Phobos, dropping a payload will have an atmospheric entry of 3 km/s, about .5 km/s less than low Mars orbit.

5800 kilometer lower Phobos tether - maximum length

Phobos orbit has an eccentricity of .0151. It bobs up and down a little. Mars' tallest mountain is about 25 kilometers tall. Given these considerations, tether can't be more than 5800 kilometers. Else the foot might crash into the top of Olympus mons.

With Zylon, tether to payload mass ratio is about 16.10.

Benefits

The tether foot will be moving about .57 km/s with regard to Mars. Mars Entry, Descent and Landing (EDL) is far simpler with .57 km/s. If Phobos is a source of propellent, much of that .57 km/s can be taken care of with reaction mass.

For an ascent vehicle, only a small suborbital hop is needed to rendezvous with the tether foot.








Wednesday, December 16, 2015

How Wolfe's tether spreadsheet works

I plan to do a series of posts examining elevators and tethers. I will link to them as posts are completed:

Lunar Elevator
Pluto Charon elevator

They will be based on Chris Wolfe's spreadsheet for modeling tethers.

I'll try to explain how Wolfe's spreadsheet works.

Tensile strength

Density and tensile strength are important quantities for tether material. Tensile strength is measured in pascals.

A pascal is a newton per square meter, newton/(meter2). A newton is a unit of force, mass times acceleration.

Zylon has a tensile strength of 580 megapascals or 580 meganewtons per square meter. On earth's surface with it's 9.8 meter/sec2 acceleration, it would take a 591,836,735 kilogram mass to exert that much force. It would take a zylon cord with a cross section of one square meter to support this force. But that's more than half a million tonnes!

10 tonnes is more plausible payload for space cargo. A much thinner cord could support this. Cross section of a Zylon cord need only be 1.72e-9 square meters. If a circular cross section, cord would be about 47 micrometers thick. Strands of hair can be anywhere from 17 to 181 micrometers thick.



So number of newtons determines tether cross sectional area.

How many newtons?

How to figure number of newtons at the tether foot? First we set maximum payload mass as well as foot station mass. The default in Wolfe's spreadsheet is a ten tonne payload mass and a foot station massing 100 kilograms. But how many newtons does this 1,100 kilogram mass exert?

The net acceleration on this foot mass is acceleration from planet's gravity minus centrifugal acceleration minus moon's gravity.

(Click on illustration to embiggen)

This spreadsheet sets the origin at the planet center.
Tether foot radius is the foot's distance from planet center.
Barycenter radius is Orbital Radius * mass planet / (mass moon/(mass planet + mass moon)
Tether anchor radius is Orbital Radius - Moon Radius. The tether anchor is assumed to be at the near point of a tide locked moon.
Distance from Barycenter to Tether Foot is Tether Food Radius - Barycenter Radius.

The three force equations:
Gravity Planet = G * Mplanet / Tether Foot Radius2
Centrifugal Accelerationω2 * Distance from Barycenter to Tether Foot. ω is constant, it is the angular velocity of the orbit.
Gravity Moon = G * Mmoon / (Orbital Radius - Tether Foot Radius)2

Net acceleration is the sum of these three.


An illustration of the accelerations with net acceleration in red. Moon gravity is negative because it is pulling away from the planet. Centrifugal acceleration is also pulling away from the planet except left of the barycenter it is towards the planet. 

When a curve crosses the axis the value is zero. Centrifugal crosses the axis at the barycenter. In most cases barycenter will be beneath planet surface. The illustration above has an exceptionally large moon. 

Net acceleration crosses the axis at L1, at this point the three accelerations sum to zero. to the right of L1, net acceleration is towards the moon.

To approximate the tether we chop it into many small lengths:


To find tether volume in step 1, we multiply the cross section by length of step 1. (Recall cross sectional area is set by number of newtons coming from tether foot.) Multiplying this volume by tether density gives step 1 tether mass. Multiplying this mass by net acceleration gives us the newtons this length exerts.

Adding the newtons from step 1 to payload newtons means the next step has a thicker cross section. We multiply this new cross section by tether length * tether density * net acceleration to get newtons from the tether length along step two.

And so on.

Summing all the masses from each step gives us total tether mass.

This is an approximation. The finer we chop the tether, the closer the approximation. The spread sheets we'll be using cut the tether length into 1,000 parts.

Our sheet can be found here. It is a 1.7 megabyte file.

For an upper moon tether, anchor will be on the far side. Moon's gravity will be added instead of subtracted from planet's gravity. I'll label tether end "Tether Top" instead of "Tether Foot".  Otherwise, the spread sheet will be the same as the lower moon tether spreadsheet.








Friday, October 30, 2015

Hope to resume space blogs soon.

I'm not dead. I've been up to my ears in alligators lately -- with paying projects (thank God!).

Hope to resume blogging soon. Some things I want to do:

Tethers and elevators

I'm eager to adapt Chris Wolfe's spreadsheet and examine a variety of tethers and elevators.

A few scenarios I want to look at:

Earth
There's a large population of dead sats in a graveyard orbit just above geosynch. These could act as a momentum bank for a vertical tether above geosynch.
I also want to look at a low earth orbit rotovator. It will be tricky adding tidal stress to the rotovator's stress from centrifugal force, but I think I can tweak Wolfe's spreadsheet to do the job.

Moon
Lunar elevator going from Mösting Crater through a balance point at EML1
Lunar vertical tether from an anchor mass at 30,000 km altitude

Mars
Phobos anchored vertical tether.
Deimos anchored vertical tether.

Ceres
Clarke style elevators from Ceres. Using Chris' spreadsheet I will be able to look at elevators of various lengths. Down the road a Ceres beanstalk might even throw stuff to trans-earth orbits.

Vesta
I want to examine Clarke style elevators from Vesta.

Boundaries of our bodies/extended phenotype

Awhile ago I reviewed a James Patrick Kelly story where a large fraction of the populace dwells in cyber-space. A trend in science fiction was been to explore artificial digital worlds rather than outer space. I opined that as telepresence improves, robotic avatars will become common place. The line between digital existence and meat space will blur.

Well, recently Kelly wrote an essay on prosthetics. It was a rich source of information, full of great web links (the norm for Kelly's Asimov articles). I want to talk about Kelly's essay. Also prosthetics and boundaries of our body. Many already regard dentures or lens implants as part of ourselves. I believe the same will become true of robotic arms and other body parts. And if we come to regard a prosthetic arm as an extension of our body, what is the difference between an arm attached to our shoulder or a robotic tele-arm thousands of kilometers away?

Robert Reed has written science fiction stories of god like beings whose minds and bodies extend throughout multiple star systems. That's not going to happen any time soon, but I do hope to see our "bodies" extend to the moon and near earth asteroids. In my lifetime.

I believe it will be advances in robotics that enable us to move beyond Cradle Earth.