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.




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.