1. Introduction

Proposals for a momentum transfer based launch system are not new. Konstantin Tsiolkovsky, credited with the concept of multi-stage rocket vehicles, also proposed the orbital tower. Much later Yuri Artsutanov inverted this idea to suggest a geostationary satellite with a counterweight and a tether extending to the Earth's surface. This so called 'space elevator' was first published in 1960 in Komsomolskaya Pravda and later discovered independently in the US when the term 'skyhook' was coined [1]. The structure was shown to be stable against the effects of lunar tidal forces and payload motions, and functions by extracting energy from Earth rotation [2]. The problem is that no known material has sufficient strength to construct a space elevator in Earth orbit.

Difficulties with the space elevator led to the proposal of the asynchronous orbital skyhook [3]. (The original concept was credited to John McCarthy at Stanford.) This is an extended orbital structure that rotates so that each end periodically comes to a low altitude and velocity, at which instants the system is easy to access. Initial studies advocated configurations that place a low demand on the tether material properties, as this was thought to be the principal challenge. To replace energy lost during launch it was proposed that the skyhook be used to return a similar quantity of material from orbit to Earth.

Detailed studies of the asynchronous skyhook [4, 5] addressed engineering aspects of the tether and docking mechanism. They proposed a set of configurations in which access is provided by a hypersonic vehicle operating at a speed of at least

3.1 km/s (Mach 10). This high speed of the access vehicle reduces the skyhook rotation rate and so places less stress on the tether material.

vM ¼ RΩ � Lω � 465 (1)

h i <sup>þ</sup> <sup>L</sup>ω<sup>2</sup> (2)

aM <sup>¼</sup> <sup>R</sup>Ω<sup>2</sup> ð Þ <sup>1</sup> � <sup>L</sup>=<sup>R</sup> �<sup>2</sup> � <sup>1</sup>

Space Access for Future Planetary Science Missions DOI: http://dx.doi.org/10.5772/intechopen.88530

rotation.

access vehicles.

negligible.

3. Mass properties

solved for the tether cross-section:

Here <sup>χ</sup><sup>2</sup> <sup>¼</sup> ρω<sup>2</sup>L<sup>2</sup>

55

Here L and ω are the skyhook half-length and rotation frequency, and Eq. 2 includes the acceleration components due to gravity, orbital velocity and skyhook

Specifying the endpoint velocity and acceleration yields two implicit equations for the skyhook parameters. With a nominal orbital radius of 8000 km the skyhook length is small enough to apply the limit L ≪ R: Then for a minimum energy state at zero velocity and 40 m/s2 acceleration, the skyhook parameters are <sup>L</sup> <sup>¼</sup> 1090 km and <sup>ω</sup> <sup>¼</sup> <sup>0</sup>:006s�1. This system can be accessed at zero velocity by a vehicle capable of ascending to an altitude of 532 km. Moreover, the maximum acceleration experienced during launch is similar to that of a conventional launch vehicle. One of the skyhook endpoints is at minimum energy when the structure is oriented radially. This state occurs with a period τ ¼ π=ð Þ ω � Ω corresponding to a ground track distance of 3176 km around the equator. The orbital parameters could be adjusted so this distance is an exact fraction of the equatorial circumference, in which case the minimum energy states occur above fixed points on the equator. These locations are natural sites at which to establish bases to operate the suborbital

The skyhook configurations of interest here have an endpoint speed near orbital

Consider a symmetric skyhook comprising two equal masses m connected by a massive tether of length 2L and define the origin at the center. The tether crosssection is a rð Þ and the tether material has uniform density ρ and ultimate tensile strength T. For a skyhook with rotation frequency ω the tension σ at radius r obeys:

ð Þ¼� <sup>r</sup> ρω<sup>2</sup>

Substituting a rð Þ¼ <sup>σ</sup>ð Þ<sup>r</sup> <sup>=</sup><sup>T</sup> and noting that a Lð Þ¼ mLω<sup>2</sup>=<sup>T</sup> this equation can be

<sup>T</sup> exp <sup>χ</sup><sup>2</sup> <sup>1</sup> � <sup>r</sup>

By symmetry the mass centroid is at the origin. This structure may be generalized to describe a set of asymmetric configurations with unequal end masses at different distances from the centroid. The symmetric configuration has the benefit of offering two opportunities to access the skyhook in each rotation cycle, but asymmetric

> ðL 0

L

=2T is a dimensionless parameter characterizing the skyhook.

ra rð Þ (3)

� �<sup>2</sup> � � � � (4)

a rð Þdr (5)

velocity to allow access at low energy. The high rotation rate means tension is mainly due to centripetal force, with the field gradient contribution being

σ0

configurations allow access to a greater variety of launch trajectories. The tether mass MT and moment of inertia IT are given by:

MT ¼ 2ρ

a rð Þ¼ mLω<sup>2</sup>

Hypersonic flight technology is not yet capable of providing routine access to the high Mach number regime. By contrast, several reusable vehicles are available that provide access to suborbital flight trajectories using combinations of air-breathing and rocket propulsion [6, 7]. High strength fiber technology has also made substantial progress with the incorporation of carbon nanotubes into the molecular structure [8]. This suggests a need to review the orbital skyhook concept with a focus on configurations that allow low speed access. It is also necessary to explore different approaches to energy replenishment that do not require access to a repository of orbiting material.

Section 2 reviews the skyhook concept and estimates the parameters of a practical launch system. Expressions for the skyhook mass properties are obtained in Section 3 for the case where centripetal force is the dominant source of tension. The dynamics is modeled in Section 4 assuming the structure remains linear, with the tether mass properties represented by a compact object at the mass centroid. Electric propulsion is proposed as a mechanism for energy replenishment in Section 5, and the feasibility of supplying propellant for the thrusters is explored. Section 6 describes the advantages of a skyhook launch system for future planetary science missions, and Section 7 summarizes the main results.

## 2. Concept description

An orbital skyhook launch system involves three phases, each exploiting a different physical process. It begins with the delivery of a payload by suborbital vehicle. Docking occurs at one of the skyhook endpoints when it is near minimum altitude and velocity. The suborbital vehicle is required to attain only a small fraction of the energy needed for orbit, and does not need to operate in a hypersonic flight regime. It can therefore employ mature airframe and propulsion technologies, making it easier to design for efficiency and reusability.

The second phase is momentum transfer from the skyhook to the payload [9]. After docking the payload gains energy as the skyhook rotates, reaching a maximum after half a cycle. If the payload is not released energy transfers back to the skyhook in the second half of the cycle as it returns to minimum energy. By selecting when the payload is released, it can be placed into an elliptical orbit or on an escape trajectory. Note that if the payload is released at a subsequent minimum energy point, the skyhook energy and orbit are left unaffected. This means the vehicle is transported around the Earth at orbital velocity, with the only energy cost being that of gaining access to the skyhook.

In the third phase energy drawn from the skyhook during launch must be replenished. If the payload mass is small relative to the total system mass, the orbital perturbation is also small. In this case the structure remains above the atmosphere through subsequent rotations, and energy replenishment may occur over an extended period. Electric propulsion is proposed for this purpose. It provides a small thrust with a large specific impulse, and therefore high propellant efficiency. Propellant can be delivered with the payload to supply thrusters at the skyhook endpoints, but it will be shown that a better approach is to apply thrust at the skyhook mass centroid.

Of interest here are skyhook configurations that offer low speed access. Ideally the endpoint speed should match the orbital velocity relative to Earth's surface. In addition, acceleration during launch must not be excessive. For a skyhook in a circular equatorial orbit with radius R and orbital frequency Ω the endpoint ground track speed and acceleration are given by:

Space Access for Future Planetary Science Missions DOI: http://dx.doi.org/10.5772/intechopen.88530

3.1 km/s (Mach 10). This high speed of the access vehicle reduces the skyhook

Hypersonic flight technology is not yet capable of providing routine access to the high Mach number regime. By contrast, several reusable vehicles are available that provide access to suborbital flight trajectories using combinations of air-breathing and rocket propulsion [6, 7]. High strength fiber technology has also made substantial progress with the incorporation of carbon nanotubes into the molecular structure [8]. This suggests a need to review the orbital skyhook concept with a focus on configurations that allow low speed access. It is also necessary to explore different approaches to energy replenishment that do not require access to a repos-

Section 2 reviews the skyhook concept and estimates the parameters of a practi-

An orbital skyhook launch system involves three phases, each exploiting a dif-

The second phase is momentum transfer from the skyhook to the payload [9]. After docking the payload gains energy as the skyhook rotates, reaching a maximum after half a cycle. If the payload is not released energy transfers back to the skyhook in the second half of the cycle as it returns to minimum energy. By selecting when the payload is released, it can be placed into an elliptical orbit or on an escape trajectory. Note that if the payload is released at a subsequent minimum energy point, the skyhook energy and orbit are left unaffected. This means the vehicle is transported around the Earth at orbital velocity, with the only energy cost

In the third phase energy drawn from the skyhook during launch must be replenished. If the payload mass is small relative to the total system mass, the orbital perturbation is also small. In this case the structure remains above the atmosphere through subsequent rotations, and energy replenishment may occur over an extended period. Electric propulsion is proposed for this purpose. It provides a small thrust with a large specific impulse, and therefore high propellant efficiency. Propellant can be delivered with the payload to supply thrusters at the skyhook endpoints, but it will be shown that a better approach is to apply thrust at the skyhook mass centroid. Of interest here are skyhook configurations that offer low speed access. Ideally the endpoint speed should match the orbital velocity relative to Earth's surface. In addition, acceleration during launch must not be excessive. For a skyhook in a circular equatorial orbit with radius R and orbital frequency Ω the endpoint ground

ferent physical process. It begins with the delivery of a payload by suborbital vehicle. Docking occurs at one of the skyhook endpoints when it is near minimum altitude and velocity. The suborbital vehicle is required to attain only a small fraction of the energy needed for orbit, and does not need to operate in a hypersonic flight regime. It can therefore employ mature airframe and propulsion technologies,

cal launch system. Expressions for the skyhook mass properties are obtained in Section 3 for the case where centripetal force is the dominant source of tension. The dynamics is modeled in Section 4 assuming the structure remains linear, with the tether mass properties represented by a compact object at the mass centroid. Electric propulsion is proposed as a mechanism for energy replenishment in Section 5, and the feasibility of supplying propellant for the thrusters is explored. Section 6 describes the advantages of a skyhook launch system for future planetary science

rotation rate and so places less stress on the tether material.

missions, and Section 7 summarizes the main results.

making it easier to design for efficiency and reusability.

being that of gaining access to the skyhook.

track speed and acceleration are given by:

54

itory of orbiting material.

Planetology - Future Explorations

2. Concept description

$$v\_M = R\Omega - L\alpha - 4\mathfrak{G} \tag{1}$$

$$a\_M = R\Omega^2 \left[ \left( \mathbf{1} - L/R \right)^{-2} - \mathbf{1} \right] + L\alpha^2 \tag{2}$$

Here L and ω are the skyhook half-length and rotation frequency, and Eq. 2 includes the acceleration components due to gravity, orbital velocity and skyhook rotation.

Specifying the endpoint velocity and acceleration yields two implicit equations for the skyhook parameters. With a nominal orbital radius of 8000 km the skyhook length is small enough to apply the limit L ≪ R: Then for a minimum energy state at zero velocity and 40 m/s2 acceleration, the skyhook parameters are <sup>L</sup> <sup>¼</sup> 1090 km and <sup>ω</sup> <sup>¼</sup> <sup>0</sup>:006s�1. This system can be accessed at zero velocity by a vehicle capable of ascending to an altitude of 532 km. Moreover, the maximum acceleration experienced during launch is similar to that of a conventional launch vehicle.

One of the skyhook endpoints is at minimum energy when the structure is oriented radially. This state occurs with a period τ ¼ π=ð Þ ω � Ω corresponding to a ground track distance of 3176 km around the equator. The orbital parameters could be adjusted so this distance is an exact fraction of the equatorial circumference, in which case the minimum energy states occur above fixed points on the equator. These locations are natural sites at which to establish bases to operate the suborbital access vehicles.

#### 3. Mass properties

The skyhook configurations of interest here have an endpoint speed near orbital velocity to allow access at low energy. The high rotation rate means tension is mainly due to centripetal force, with the field gradient contribution being negligible.

Consider a symmetric skyhook comprising two equal masses m connected by a massive tether of length 2L and define the origin at the center. The tether crosssection is a rð Þ and the tether material has uniform density ρ and ultimate tensile strength T. For a skyhook with rotation frequency ω the tension σ at radius r obeys:

$$
\sigma'(r) = -\rho a^2 r a(r) \tag{3}
$$

Substituting a rð Þ¼ <sup>σ</sup>ð Þ<sup>r</sup> <sup>=</sup><sup>T</sup> and noting that a Lð Þ¼ mLω<sup>2</sup>=<sup>T</sup> this equation can be solved for the tether cross-section:

$$a(r) = \frac{mL\alpha^2}{T} \cdot \exp\left[\chi^2 \left\{1 - \left(\frac{r}{L}\right)^2\right\}\right] \tag{4}$$

Here <sup>χ</sup><sup>2</sup> <sup>¼</sup> ρω<sup>2</sup>L<sup>2</sup> =2T is a dimensionless parameter characterizing the skyhook. By symmetry the mass centroid is at the origin. This structure may be generalized to describe a set of asymmetric configurations with unequal end masses at different distances from the centroid. The symmetric configuration has the benefit of offering two opportunities to access the skyhook in each rotation cycle, but asymmetric configurations allow access to a greater variety of launch trajectories.

The tether mass MT and moment of inertia IT are given by:

$$M\_T = 2\rho \int\_0^L a(r) dr \tag{5}$$

$$I\_T = 2\rho \int\_0^L a(r)r^2 dr\tag{6}$$

'worst case' in the sense that skyhook rotation is specified to allow access at zero velocity relative to the Earth. If the access vehicle provides a horizontal velocity component the rotation rate is smaller, in which case the taper factor and skyhook

Skyhook length is a significant factor in the dynamics because field strength is not uniform across the structure. This differs from most problems in astrodynamics where the object of interest is small compared to the field gradient length scale, or

Here the skyhook is assumed to behave as a rigid body, kept in tension by the rotation and experiencing no stretching or bending. The validity of these assumptions depends on the tether material properties, but they are sufficient for the present purpose. The structure is expected to remain linear due to the large

The equations of motion of a rigid body are typically obtained by a Lagrangian

The skyhook system is modeled here as three objects connected by tethers of fixed length L as illustrated in Figure 1. The central object has the mass properties of the tethers as calculated above. This formulation represents the physical extent of the skyhook in a non-uniform field. It is also a good approximation for the mass distribution of the tether if it has a significant taper factor, in which case much of the mass is concentrated near the centroid. Based on these considerations a Newto-

The system state is described by a six element vector comprising the skyhook centroid location r ¼ ð Þ r; θ and orientation angle φ and their derivatives. The endpoint locations are specified by the vectors r1 and r2 which are functions of the state

Skyhook geometry with the tether mass and moment of inertia represented by a compact object at the mass

method using the mass properties. This formulation ignores the field gradient effect, which is important for skyhook dynamics. To see this note that the skyhook structure experiences a moment due to the two arms being subject to different field strengths according to their proximity to Earth. If the skyhook were treated as a

the system can be simplified by assuming spherical symmetry.

single compact object this behavior would not be represented.

centripetal restoring force that counters any bending.

nian formulation is used for the analysis.

Figure 1.

centroid.

57

mass are also decreased.

Space Access for Future Planetary Science Missions DOI: http://dx.doi.org/10.5772/intechopen.88530

4. Equations of motion

Evaluating the integrals and simplifying:

$$\mathbf{M}\_{\mathrm{T}}/\mathfrak{m} = 2\sqrt{\pi} \,\,\chi\,\exp\left[\chi^2\right] \mathbf{erf}(\chi) \tag{7}$$

$$\mathrm{I}\_{T}/\mathfrak{mL}^{2} = \sqrt{\pi} \chi^{-1} \exp\left[\chi^{2}\right] \mathrm{erf}(\chi) - 2 \tag{8}$$

The limit χ ! 0 represents a material of infinite strength, in which case the tether mass and moment of inertia vanish. Adding the contributions of the two end masses leads to expressions for the mass properties of the entire skyhook:

$$M/m = 2\sqrt{\pi} \,\chi\exp\left[\chi^2\right] \,\text{erf}(\chi) + 2\tag{9}$$

$$\text{I/mL}^2 = \sqrt{\pi} \,\chi^{-1} \exp\left[\chi^2\right] \text{erf}(\chi) \tag{10}$$

These expressions for the skyhook mass properties indicate the dependence on tether material properties, and provide key parameters for dynamical modeling.

An important feature of a tether is the taper factor, the ratio of maximum to minimum cross-section area. A tether constructed from low strength material has a large taper factor, indicating its impracticality. The nominal skyhook described above with a carbon fiber tether has a taper factor of 237, in which case the diameter at the centroid is about 15 times that the end points. If the tether had the properties of carbon nanotubes the taper factor reduces to 3.3. The properties of any future tether material are likely to fall within these bounds.

Table 1 indicates the mass properties of the nominal skyhook for several tether materials. Notionally high strength materials like steel and diamond are excluded by the very large taper factor. Aramid fibers like Kevlar are possible but the total mass is large. The strongest carbon fiber offers a solution with a skyhook mass about 4600 times the endpoint mass. If materials with still greater tensile strength become available, such as by incorporating carbon nanotubes or colossal carbon tubes into the tether material, the taper factor and skyhook mass can be much smaller.

For the skyhook configuration described here the endpoint mass is regarded as the maximum payload capability. This assumes the endpoint mass may be replaced by a docking mechanism of negligible mass to capture the payload. Engineering margins have not been included in this analysis, but the nominal configuration is a


#### Table 1.

Tether mass properties for various materials (from Eqs. (4), (7) and (8)).

'worst case' in the sense that skyhook rotation is specified to allow access at zero velocity relative to the Earth. If the access vehicle provides a horizontal velocity component the rotation rate is smaller, in which case the taper factor and skyhook mass are also decreased.
