rotovator release orbits.cdf

Reaching Circular Orbits Using a Rotovator
Gerry Flanagan
April 22, 2011, Update July 21 2011.
Introduction
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 DV required.
The first component is the velocity need to hook up to the rotovator, and the second component is the DV 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
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rotovator release orbits.cdf
Studies
DV Requirements and Access able Orbit Altitudes for Α=0.5
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 DV 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 DV 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 DV 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 DV 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.
Delta V, msec
10 000
8000
6000
4000
2000
0
1000
2000
Orbit Height, km
4000
3000
Figure 2. Rocket DV 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 DV over the DV 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.
Delta V Fraction
1.0
0.8
0.6
0.4
0.2
0.0
0
500
1000
1500
2000
2500
Orbit Height, km
3000
Figure 3. Fraction of DV 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.
rotovator release orbits.cdf
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Delta V Fraction
1.0
0.8
0.6
0.4
0.2
Orbit Height, km
0.0
0
10 000
20 000
30 000
40 000
Figure 4. Fraction of DV 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.
Total DV requirement versus the parameter Α
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 DV 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 DV includes the initial launch to rotovator docking,
and the DV needed to circularize the orbit. The following plots examine the total DV 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 DV. 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.
Delta V Fraction
0.8
Α=0.3
Α=0.5
0.6
Α=0.7
0.4
0.2
0.0
Orbit Height, km
0
200
400
600
800
1000
1200
1400
Figure 5. DV fraction relative to pure rocket launch. Orbits are circularized from release ellipse perigee. Assumed parameters; 200 km capture altitude,
1500 km tether.
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rotovator release orbits.cdf
Delta V Fraction
0.8
Α=0.3
Α=0.5
0.6
Α=0.7
0.4
0.2
Orbit Height, km
10 000
20 000
30 000
40 000
Figure 6. DV fraction relative to pure rocket launch. Orbits are circularized from release ellipse apogee. Assumed parameters; 200 km capture altitude,
1500 km tether.
Examine the release angles for Α=0.5
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.
Orbit Height, km
1400
1200
1000
800
600
400
200
Release Angle
0
0
50
100
150
200
250
300
350
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°.
rotovator release orbits.cdf
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Orbit Height, km
40 000
30 000
20 000
10 000
Release Angle
0
50
100
150
200
250
300
350
Figure 8. Orbit height up to geosynchronous versus release angle. Assumed parameters; 200 km capture altitude, 1500 km tether and Α=0.5
Graphical Representation of orbits for a range of release angles.
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.
Release Angle=30
A 90° angle would be a typical low orbit insertion. The lifter would perform a rocket burn at the perigee of the ellipse.
Release Angle=90
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rotovator release orbits.cdf
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°.
Release Angle=120
Releases at greater than 180° are feasible, but I haven’t found any case where it would be advantageous.
Release Angle=260
rotovator release orbits.cdf
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Ultimately, you just slam the payload back into the earth.
Release Angle=300
A Manipulate window for viewing release orbits
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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rotovator release orbits.cdf
Out[41]=
ReleaseAngle
PickupAltitude
TetherLength
PickupVelRatio
scale
PickupVelRatio=0.5
TetherLength=1200km
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