Momentum Exchange Tethers — Early History

The history of momentum-exchange tethers goes back many, many years but is bound by a common thread that, until recently, limited the realization of this technology. That common thread is the need for high specific tensile strength.

The first idea of concept of a tether dates back to the imagination of Konstantin Tsiolkovsky, the Russian schoolteacher who first developed our modern concepts of rocketry and first derived the rocket equation. In the late 1800s, Tsiolkovsky visited Paris and saw the Eiffel Tower. He was so impressed by the sight that he imagined a tower reaching up far into space. He calculated the height at which such a tower would have to be before the centrifugal force from the Earth’s rotation balanced the pull of gravity (inadvertently calculating the altitude of geosynchronous orbit).

Tsiolkovsky, of course, could not conceive of any material that could withstand the compressive forces of such a structure, but sixty years later, a Russian engineer named Yuri Artsutanov picked up the thread of Tsiolkovsky’s work and first worked out the engineering principles of what is now called a “space elevator”, a long tether hanging all the way from geosynchronous orbit to the surface of the Earth. The space elevator required materials with specific tensile strength far in excess of any known material, and still does. Further conceptual engineering work on the space elevator concept was done in the early 1970’s by American engineer Jerome Pearson.

The space elevator was a hanging tether, and payloads were required to traverse its length in order to achieve orbit. The beginnings of rotating momentum-exchange tethers date to the late 1970s, when Hans Moravec, a robotics researcher at Stanford University (now at Carnegie-Mellon) was intrigued by a suggestion of his friend John McCarthy of a satellite that “rolled like a wheel” around the Earth. Moravec began a scientific investigation of the concept, which he first called a “non-synchronous orbital skyhook” and later a “Rotovator”. Like the space elevator, it reached all the way to the surface of the Earth, but unlike the elevator, it rotated about its axis a number of times per orbit. A payload would be picked up by the tip at the surface of the Earth and then thrown half a rotation later into an interplanetary trajectory. The Rotovator was a good deal shorter than the space elevator (~4200 km vs. 40,000-100,000 km) but was not much better in terms of materials required. Moravec published a paper on the subject in the Journal of Astronautical Sciences where he speculated on advanced forms of matter that might have the strength needed to build the Rotovator.

About a year after the JAS paper was published, Dupont’s development of Kevlar excited Moravec to the possibilities of Rotovators built with conventional materials. He wrote a short paper called on the subject which was never published. The paper showed that Kevlar skyhooks were not feasible around the Earth but could be reasonably built around the Moon and Mars. In an appendix to this unpublished paper, Moravec speculated on the possibility of skyhooks built in interplanetary space that would assist spacecraft traveling between the Earth and Mars. To the great benefit of future tether researchers, his equations for the cross-section of a tether, in the absence of a gravitational field, could be integrated in closed-form. Thus, the Moravec “tether equation” was first derived.

(1)    \begin{equation*} v_c = \sqrt{\frac{2 \tau}{f \rho}} \end{equation*}

     \begin{displaymath} \chi = \frac{v_\text{tip}}{v_c} = \sqrt{\frac{v^2_\text{tip} f \rho}{2\tau}} \end{displaymath}

     \begin{displaymath} A(r) = \left(\frac{m_{tip}\:v_{tip}^2\:f}{\ell \tau}\right) \exp \left(\chi^2 \left(1 - \frac{r^2}{\ell^2}\right) \right) \end{displaymath}

Moravec was able to derive analytical expressions for the area of the tether as a function of its distance from the rotational center. He then numerically integrated the area expression along the length of the tether to calculate volume and mass. As an aside, in an appendix, he considered the case of a tether spinning in free space. When the tension on the tether was only due to centrifugal forces, the area expression could be analytically integrated to a closed-form solution. Thus the Moravec mass ratio was derived.

     \begin{displaymath} A(r) = \left(\frac{M_\text{tip} v^2_\text{tip} f}{\ell \tau}\right) \exp \left(\frac{f\rho}{\tau}\frac{v^2_\text{tip}}{2}\left(1 - \frac{r^2}{\ell^2}\right)\right) \end{displaymath}

     \begin{displaymath} m_\text{tether} = \rho\int^\ell_0 A(r) dr \end{displaymath}

The equation, once integrated, includes the Gaussian error function “erf”.

     \begin{displaymath} \frac{m_\text{tether}}{m_\text{tip}} = \sqrt{\pi \frac{f\rho}{\tau}\frac{v^2_\text{tip}}{2}} \exp\left(\frac{f\rho}{\tau}\frac{v^2_\text{tip}}{2}\right) \text{erf}\left(\sqrt{\frac{f\rho}{\tau}\frac{v^2_\text{tip}}{2}}\right) \end{displaymath}

The equation could be simplified by realizing that fundamentally, the mass ratio is a function only of the velocity ratio, which itself is the ratio of the tip velocity of the tether and the characteristic velocity of the tether material.

     \begin{displaymath} \frac{m_\text{tether}}{m_\text{tip}} = \sqrt{\pi \frac{v^2_\text{tip}}{v^2_c}} \exp\left(\frac{v_\text{tip}}{v_c}\right)^2 \operatorname{erf}\left(\sqrt{\frac{v^2_\text{tip}}{v^2_c}}\right) \end{displaymath}

     \begin{displaymath} \frac{m_\text{tether}}{m_\text{tip}} = \sqrt{\pi} \left(\frac{v_\text{tip}}{v_c}}\right) \exp\left(\frac{v_\text{tip}}{v_c}\right)^2 \operatorname{erf}\left(\frac{v_\text{tip}}{v_c}\right) \end{displaymath}

In a fascinating simplification, the mass ratio of the tether turns out to be only dependent on a single value, the velocity ratio between the tip velocity of the tether and the characteristic velocity of the tether’s material. We denote this velocity ratio with the Greek letter “chi”.

     \begin{displaymath} MR = \frac{m_\text{tether}}{m_\text{tip}} = \sqrt{\pi}(\chi) \exp(\chi^2) \operatorname{erf}(\chi) \end{displaymath}

Further insight into the value of the equation was gained by comparing it to the rocket equation and noting the similarities and differences.

Moravec wrote a few articles on the subject for space-themed publications, but basically returned to his robotics work. Nevertheless, Moravec’s equation still serves as a foundation to all momentum-exchange tether work to this day.

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MS, nuclear engineering, University of Tennessee, 2014, Flibe Energy, president, 2011-present, Teledyne Brown Engineering, chief nuclear technologist, 2010-2011, NASA Marshall Space Flight Center, aerospace engineer, 2000-2010, MS, aerospace engineering, Georgia Tech, 1999

About Kirk Sorensen

MS, nuclear engineering, University of Tennessee, 2014, Flibe Energy, president, 2011-present, Teledyne Brown Engineering, chief nuclear technologist, 2010-2011, NASA Marshall Space Flight Center, aerospace engineer, 2000-2010, MS, aerospace engineering, Georgia Tech, 1999
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5 Responses to Momentum Exchange Tethers — Early History

  1. I want to express my gratitude to the authors of this article for indirectly pointing out that other technological concepts for the construction of Space Elevators have been proposed.

    I am a believer that a functional space elevator would only be possible in a hybrid form, i.e. by a combination of different technologies.

    Factors limiting the construction/deployment of a space elevator include among others the possibility that, due to unforeseeable reasons, the popular CNT tether may not be achievable after all, (perhaps due to material properties,) material deterioration due to external agents, (such as meteorites,) or perhaps we may not have the means for the foundations of the underlining requirements such as that of a asteroid relocation. So we need a plan B to back up our ambitions.

    Consider our proposal at

    Best Regards

  2. Chris (Robotbeat) says:

    This isn’t so bad. Current state-of-the-art material properties (i.e. ~4GPa strength, 1.3kg/liter density) would allow a 3 or 4 km/s tether without too insane of a mass ratio (16 or 100, respectively), which certainly could cut down on the 8 or 9km/s a spacecraft usually needs to achieve orbit. You need a massive tether anyways, so your tether doesn’t get dragged down into the atmosphere by your payload.

    By using a tether, your first-stage delta-v requirements are now within spitting distance of an air-breathing scramjet, although just a single-stage rocket of some sort will probably end up being cheaper with fewer operational constraints and far lower development costs. A regular kerolox first stage could probably do it easily enough, but a kerohydrolox thrust-augmented nozzle first stage would probably give you more margin, perhaps allowing a reusable VTVL rocket.

    The point is to find better ways to break up the exponent and improve possible Isp. You don’t need one part of your orbital system doing all the work.

  3. Chris (Robotbeat) says:

    Just kidding, spectra has better specific strength than that, actually. The mass ratios for a 3km/s and 4km/s tether would be 6 and 23, respectively. Much better.

  4. Chris (Robotbeat) says:

    It looks like the domain has expired or something! Most of the images are broken links!

  5. Chris (Robotbeat) says:

    This is pretty frustrating. The images don’t even show up in (perhaps you put in your robots.txt file for not to archive your page… bad decision, in case it ever expires!).

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