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:

LEO Rotovator
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.

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