Reaching Circular Orbits Using a Rotovator

Gerry Flanagan

April 22, 2011, Update July 21 2011.

A rotovator is a rotating, orbiting tether that captures a payload (we’ll call a “lifter”), carries the lifter around a portion of the tethers arc, and then releases it along a new trajectory. Figure 1 shows the basic arrangement. The operation typically consider for a rotovator assumes that the payload release occurs when the cable reaches the highest point, 180° around the circle. This position gives the maximum velocity gain to the payload. Because our current interest is in rotovators that allow for a substantial reduction in the initial launch velocity as compared to LEO velocity, a 180° position results in the payload exceeding escape velocity. This may be the desired affect, but I’d like to examine using a rotovator for efficiently reaching a circular earth orbit. The assumption is that the majority of mass going into space will go into near circular orbits, and the large mass flow into circular orbit will justify the cost of the rotovator. The easiest way to reach a range of orbits is to vary the release point. For a range of release points, the payload will go into an elliptical orbit. At either the perigee or apogee of the ellipse, and additional rocket burn would be used to circularize the orbit. The efficiency of the system can be measured by the total rocket ΔV required. The first component is the velocity need to hook up to the rotovator, and the second component is the ΔV need to circularize at the desired altitude.

The solutions use closed-form two-body elliptic orbit equations. We do not integrate the equations of motion. The basic idea is to find the elliptical orbit that has the same velocity and position vector as the release point. Finding the orbit with the same conditions as the release involves a simple numerical search. The elliptical orbit after the release is circularized by a rocket burn at either the ellipse apogee or perigee. Depending on the initial conditions (for example, a release point that is great than 180° around the tether rotation), the apogee may be below the surface of the earth, making that option impossible.

For this study, we will not explicitly consider the tether mass given realistic materials. Some of the original work on this system considered a tip velocity that was zero relative to the earth’s surface (imagine a wheel rolling around the earth). For this rotational speed, the materials issue is challenging and the tether mass is excessive compared to the payload mass. However, by allowing for some relative velocity with the earth’s surface, reasonable tether-to-payload mass ratios can be achieved. For this study, we focus on a tip velocity that is 50% of the orbital velocity at the capture altitude.

Figure 1. Schematic of rotovator and definitions of

An important parameter is the velocity of the tether tip relative to the earth surface. This factor will determine the propellant mass fraction of the lifter rocket that takes a payload from the surface to meet with the tip (or, the speed requirement for a hypersonic air-breathing vehicle). We are characterizing this speed with the parameter α which is defined as the rendezvous speed over the velocity for a circular orbit at the altitude of the rendezvous. Thus, a small α indicates a smaller ΔV requirement for the lifter rocket. The trade-off is that the cable mass (relative to the payload mass) will grow exponentially as α decreases and may be unacceptably large for α=0. Also, one should consider the total ΔV requirement to reach the desired angle. This includes the velocity increment needed to rendezvous with the tether along with the rocket burn needed to circularize the orbit after tether release.

First, we will consider the rocket ΔV needed to reach various circular orbits altitudes given a rotovator assist. The calculation approach will be to determine the resulting elliptical orbit that results after releasing the lifter at a particular angle around the tethers path. The orbits can be circularized at either the apogee or perigee of the ellipse, giving two possible circular orbits for any release point. Some of the perigee orbits are not possible because the ellipse hits the earth before the perigee point. As an example, Figure 2 shows the total ΔV needed to reach a given orbit altitudes for α=0.5, a tether length (from the CG to tip) of 1500 km, and a pickup point that is 200 km above the earth.

Figure 2. Rocket ΔV need to circularize orbit for 200 km capture altitude, 1500 km tether, and rendezvous at 0.5 orbital velocity. Blue is for circularization from perigee, red is for circularization from apogee.

Figure 3 is the same data, except plotted as the ratio of the required lifter ΔV over the ΔV for a pure rocket launch to the same orbit. The plot shows that a wide range of orbital altitudes can be achieved for less than 60% of the velocity change needed for a pure rocket insertion. It is also notable that a single rotovator can be use for a wide range of orbits, giving the device a lot of operational flexibility. Figure 4 is an extended the range version of Figure 3, with the orbit height going to the equivalent of geosynchronous.

Figure 3. Fraction of ΔV need to circularize orbit relative to pure rocket launch. Assumed parameters; 200 km capture altitude, 1500 km tether, and rendezvous at 0.5 orbital velocity. Blue is for circularization from perigee, red is for circularization from apogee.

Figure 4. Fraction of ΔV need to circularize orbit relative to pure rocket launch. Assumed parameters; 200 km capture altitude, 1500 km tether, and rendezvous at 0.5 orbital velocity. Blue is for circularization from perigee, red is for circularization from apogee.

α is the ratio of the capture velocity (relative to earth) over the orbital velocity at the capture altitude. It is a measure of the ΔV saved by the rocket for the initial boost. Small values of α should result in a higher payload fraction at final orbit, but require a more massive cable. An α of less than about 0.4 may not be possible with existing engineering materials. The total ΔV includes the initial launch to rotovator docking, and the ΔV needed to circularize the orbit. The following plots examine the total ΔV for a series of α values. Figure 5 shows the results for cicularization from the perigee. From the results in the previous section, this is the preferred operation for orbits lower than the rotovator CG. As expected, a lower α results in a lower total ΔV. However, the difference between α=0.3 and α=0.5 is modest. Figure 6 is a similar plot for cicularization from the apogee. This is the preferred operation for orbital altitudes greater than the rotovator CG. For high altitude orbits, the advantage of the smaller α is more significant.

Figure 5. ΔV fraction relative to pure rocket launch. Orbits are circularized from release ellipse perigee. Assumed parameters; 200 km capture altitude, 1500 km tether.

Figure 6. ΔV fraction relative to pure rocket launch. Orbits are circularized from release ellipse apogee. Assumed parameters; 200 km capture altitude, 1500 km tether.

The presentations in the figures above regarding deltaV possibilities doe not show the release angles needed to achieve a given orbit. We’ll repeat the calculations for α=0.5 and replot. Figure 7 shows the orbit altitude versus the release angle. As before blue dots are for circularization at the perigee, and the red dots are for circularization at the apogee. Figure 7 has been scaled to focus on low earth orbits.

Figure 7. Orbit height versus release angle. Assumed parameters; 200 km capture altitude, 1500 km tether and α=0.5

For the lower altitudes, we know that circularizing the perigee is more efficient. We can see from Figure 7 that the full range of altitudes (up to 1500 km) can be achieved using a very narrow range of release angles, going from approximately 80° to 100°. One could also use a range of angles going from 260° to 280°, but there doesn’t seem to be any advantage to every going beyond 180°. Figure 8 is the same data but plotted to show a wider range of orbital heights. For higher altitudes, including geosynchronous orbit, one uses the apogee with angles ranging from around 70° to 160°.

Figure 8. Orbit height up to geosynchronous versus release angle. Assumed parameters; 200 km capture altitude, 1500 km tether and α=0.5

The following graphics are scaled images of the lifter orbit after release for selected release angles. Note that the graphic shows the rotovator as symmetric about the CG. The input parameters are pickup altitude of 200 km, tether length of 1500 km, α=0.5.

For small angles (<80), the perigee hits the earth and is useless. The apogee could be used, but the deltaV calculations show that this would be inefficient.

A 90° angle would be a typical low orbit insertion. The lifter would perform a rocket burn at the perigee of the ellipse.

Increasing the release angle increases the apogee altitude. For high altitude orbits, the circularization would occur at the apogee. Escape velocity is achieved at approximately 140°.

Releases at greater than 180° are feasible, but I haven’t found any case where it would be advantageous.

Ultimately, you just slam the payload back into the earth.

The manipulate shows a to-scale image of the elliptical orbit for a specified set of release parameters. The image shows the rotovator at the moment of release. In the manipulate controls, the parameter PickupVelRatio is the same as α in the text.

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