Golden Tethers

Φ,  also known as the golden ratio, is one of my favorite numbers. It is (sqrt(5) + 1) / 2, approximately 1.618. I've done many paintings and drawings using this number. Here are a couple:

 Two images from my coloring books

The number occurs naturally in designs having a 5 fold symmetry but it also turns up in unexpected places. I was happy to find it when I was playing with orbital tethers.

Vertical Tethers vs Space Elevators


Gravity gradient stabilized vertical tethers are smaller cousins of a full blown space elevator. Jerome Pearson has developed equations giving a space elevator's dimensions and taper ratio.

Some of Pearson's terms:

r₀ planet's radius
g₀ planet's surface gravity
r_s radius of planet's synchronous orbit

For looking at vertical tethers I use P. K. Aravind's equations which I believe are based on Pearson's work. But I substitute the above terms with r_f for r₀,  g_f for g₀, and  r_c for r_s.

Tether Foot
The term r_f refers to distance from planet center to tether foot. Imagine a planet the same mass of earth but with a larger radius, r_f. Then r_f and r₀ become the same. Same with surface gravity, gravity at the tether foot would be the same as surface gravity of a planet with radius r_f.

Tether Center
The term r_c is the distance from planet center to radius at which a natural circular orbit would have the same angular velocity as the tether we're looking. I call this the tether center.  Misnamed since the length above the "center" is greater than the length below, but I can't think of a better word. Again, we can imagine a planet whose angular velocity is the same as our tether's, so r_c would become  r_s.

Tether Top
The term r_t can remain the same. The tether top applies to a vertical tether just as much as it does to a space elevator. The length above the tether center must balance the length below.

Tether Size

I would like to make the tether as small as possible. Smaller size makes for less materials that have to be launched to space. A shorter length makes for greater throughput, less stress allowing less exotic tether materials and smaller taper ratios, and a smaller cross section thus reducing vulnerability to debris impacts.

The tether should be as low as possible. A lower r_c makes for a higher angular velocity and a better Oberth benefit.

How low a tether foot can descend is limited by height of atmosphere. We want the foot above the atmosphere as drag would pull the tether down. So  r_f is one of the first quantities considered in my tether spreadsheet.

How high to make r_t? If releasing a payload from tether top sends the payload on a parabolic trajectory, we can choose any apoapsis by releasing from tether locations between r_t  and r_c.

Releasing a payload from a point (1+e)^1/3 r_c will send the payload on conic section trajectory having eccentricity e. The eccentricity of a parabola is 1. So an r_t  = 2^1/3 r_c would give us a tether able to deliver payloads to any apoapsis.

Adapting P. K. Aravind's equation (5) from his The physics of the space elevator we have

(r_f / 2) * [sqrt(1 + 8(r_c/r_f)³) - 1] = r_t

Recalling we want r_t to send payloads on a parabolic path...

(r_f / 2) * [sqrt(1 + 8(r_c/r_f)³) - 1] = 2^1/3 r_c

Setting our units r_c = 1 ...

(r_f / 2) * [sqrt(1 + 8/(r_f³)) - 1] = 2^1/3

Which comes to the suprising and pleasing result...

r_t  = Φ r_f

Where Φ is the golden mean, the number I was talking about at the beginning of this blog post.

The velocity of the golden tether's foot is about 68.7% the velocity of a normal circular orbit at r_f .

Golden Earth Tether.

The top red orbit is a parabola.

The foot is 300 km above earth's surface. It's moving about 4.8 km/s wrt earth's equator.

Using Kevlar, taper ratio is about 5.1. Tether length is about 4130 km.

Golden Moon Tether.

The top red orbit is a parabola.

The foot is 80 km above moon's surface. It's moving about 1.13 km/s wrt moon's surface.

Using Kevlar, taper ratio is about 1.1. Tether length is about 1125 km.

The tether doesn't have to be golden. Longer tethers would be able to send payloads on hyperbolic orbits (e > 1), useful if interplanetary Hohmann transfers are desired. Shorter tethers would be limited to elliptical orbits (e < 1), but this could still be useful. This spreadsheet allows the user to set eccentricity of exit orbit as well as body's mass and radius. You can also set the altitude of tether foot.