Human Space Exploration in the “Deep Space Proving Grounds”

Abstract

Freeman Dyson once said “there is nothing so great or so crazy that a technological society cannot be moved to do, provided that it is physically possible.” He is one of the few surviving giants among visionaries from the space race of the 1960s. In his book Disturbing the Universe, he presents heretical ideas, such as space colonisation and the exploitation of natural resources in space. He examines two scenarios. The first one explores the feasibility of a habitat around the Sun, which uses its energy to sustain life. Public money would finance this project. The second scenario analyses the activities of small groups of settlers, who use the resources of asteroids to support their operations. His preferences lie with the latter scenario:

Appendix: Technical Notes on the Net Present Value

This appendix focuses on the determinants of the hypothetical case for net present value:

$$ \mathrm{N}\mathrm{P}\mathrm{V}=\left[{\displaystyle \sum}\left({p}_t\times \kern0.28em {q}_t-\kern0.28em {\mathrm{oc}}_t\times \kern0.28em {q}_t\right)\times {e}^{-\delta t}\right]-C $$

Following Sonter, the volume of metals brought back from the asteroid to LEO (∑ q _t ) are derived from the Tsiolkovsky rocket equation, which relates Δv with the effective exhaust velocity and the initial and final masses of a rocket. As the space vehicle’s mass is small compared to the mass of the metals, the latter is equal, with a certain approximation, to:

$$ {\displaystyle \sum }{q}_t={M}_{\mathrm{produced}}\times {e}^{-\varDelta v/{v}_e} $$

(3.4)

where M _produced is the mass produced on the asteroid, Δv is the change in velocity required for the return by orbital transfer, and v _e is the exhaust velocity.

The mass produced on the asteroid is equal to:

$$ {M}_{\mathrm{produced}}={M}_{\mathrm{mpe}}\times \kern0.28em f\times \kern0.28em {t}_{\mathrm{stm}} $$

(3.5)

where M _mpe is the mass of the equipment, f is the throughput factor expressed as the ratio of kilograms per day of the materials treated per kilogram of equipment, and t _stm is the time of stay on the asteroid (mining season). Substituting (3.5) into (3.4) gives:

$$ {\displaystyle \sum }{q}_t=\left({M}_{\mathrm{mpe}}\times f\times {t}_{\mathrm{stm}}\right)\times {e}^{-\varDelta v/{v}_e} $$

(3.6)

This equation shows the mass of the metals returned to LEO and depends on technical factors, the length of the season, the propulsion system adopted, and the change in velocity necessary for the transfer between the two orbits.

It remains to find Δv, the change in velocity needed for the return trip. Δv is a scalar quantity dependent on the desired trajectory and not on the mass of the space vehicle. It is calculated as the sum of the Δv’s required for the propulsive manoeuvres during the mission and determines how much propellant is required for a vehicle of a given mass and propulsion system. The Δv _r to insert into the return trajectory is small, due to the low gravity of NEAs, but Δv _ic to correct for inclination change must be added.

The asteroid in the hypothetical case is an Aten type. This mission profile assumes a target body with a semi-major axis less than 1, a quasi-circular orbit (an Aten), and a mining season commencing at perihelion and running until aphelion. This mission assumes a Hohmann transfer to rendezvous with the target asteroid at its perihelion and with a near-aphelion departure after half an orbit stay time. The Hohmann transfer orbit is elliptical and used to transfer objects between two circular orbits of different radii in the same plane. An example of the Hohmann transfer is illustrated by Fig. 3.4.

Fig. 3.4

Mission to an Aten asteroid

Following Hohmann,²⁸ the minimum change in velocity for the return trip is equal to the difference between the required velocity to enter the transfer orbit (v ₁) and the transfer ellipse velocity (v ₂):

$$ \varDelta {v}_r=\left({v}_1-{v}_2\right) $$

(3.7)

where:

$$ {v}_1=\kern0.28em \surd \left(\mu /\left(\left(1.5\times {10}_{11}\right)\times Q\right)\times \left(1-{e}_{\mathrm{transfer}}\right)\right) $$

(3.8)

and

$$ {v}_2=\surd \left(\mu /\left(\left(1.5\kern0.5em \times \kern0.5em {10}_{11}\right)\times Q\right)\right)\times \left(1-{e}_{\mathrm{tr}}\right) $$

(3.9)

where:

• μ = G × m _s ,

• G = Newton’s gravitational constant,

• m _s  = mass of the Sun,

• Q = the aphelion of the asteroid in AU,

• e _transfer = (1 − q)/(1 + q),

• q = the perihelion of the asteroid in AU,

• e _tr = the eccentricity of the asteroid.

Using the orbital characteristics of asteroid 2004 MN4, v ₁ is equal to 37.62 km/s; v ₂ is equal to 36.9 km/s. Thus the change in velocity for transfer orbit insertion is 0.73 km/s. However, we have to add the velocity change required for the inclination change, which depends on where the change occurs and the heliocentric velocity at that point. This velocity change can be approximated by:

$$ \varDelta {v}_{\mathrm{ic}}=\left(0.52\kern0.5em \times \kern0.5em \mathrm{inclination}\right) $$

(3.10)

which, in the hypothetical case, is equal to 1.73 km/s. The total velocity change for the return to Earth transfer is then equal to:

$$ \varDelta {v}_T=\varDelta {v}_r+\kern0.28em \varDelta {v}_{\mathrm{ic}} $$

(3.11)

or 2.46 km/s.

We assume that the fuel needed for the return trip is extracted from the asteroid. Let us assume the use of a nuclear thermal rocket, which were being actively developed under the NERVA project. Conceptually, this works by pumping hydrogen through the core of a nuclear reactor to heat it to produce exhaust as a reaction fluid. A full-scale operating nuclear rocket was tested and achieved a specific impulse (Isp) of 850 seconds and nearly attained flight readiness before funding was withdrawn. We assume that such a rocket achieves an Isp of 850 seconds. The exhaust velocity is around 8.3 km/s. as the exhaust velocity is equal to the specific impulse times the standard acceleration of gravity. If one assumes that the mass of the spaceship is negligible with respect to the mass of the payload, from the rocket equation, the following holds:

$$ {M}_{\mathrm{start}}={M}_{\mathrm{finish}}\kern0.5em \times \kern0.28em {e}^{\varDelta v/{v}_e} $$

(3.12)

In our hypothetical case, M _finish is about 800 tonnes and ${e}^{\varDelta v/{v}_e}$ is 1.36. The amount of fuel needed for the return trip is about 320 tonnes. The total time of the mission is equal to the time needed for research and development and the construction of the equipment and the spaceship (t₁), plus the transfer time from LEO to the asteroid and return to LEO. We assume that t ₁ is about 1.1 years, and the time from LEO to the asteroid and return to LEO is approximately equal to 3T/2, where T is the orbital period of the asteroid. In our hypothetical case, this is equal to 1.3 years and the total time of the mission is 2.4 years.

We are finally able to rewrite the equation for the net present value of the project as:

$$ \begin{array}{l}\mathrm{N}\mathrm{P}\mathrm{V}=\Big[{{\displaystyle \sum n}}_{t=t1+3T/2}\left(\left({p}_t\kern0.5em -\kern0.5em {\mathrm{oc}}_t\right)\kern0.5em \times \kern0.5em {\left(\left(\left({M}_{\mathrm{mpe}}\times \kern0.28em f\kern0.28em \times \kern0.28em {t}_{\mathrm{stm}}\right)\kern0.5em \times \kern0.5em {e}^{-\varDelta {v}_{return}/{v}_e}\right)/n\right)}_t\right)\\ {}\kern3.75em \times \kern0.5em {e^{-\delta}}^t\Big]-C\end{array} $$

(3.13)

under the assumption that the investment takes place in year 0, and of constant sales for n years.

Equation (3.13) shows that the total mission time is crucial to the feasibility of the project. The longer the mission time is, the lower the net present value of the project. Δv for rendezvous with the asteroid enters into this equation through C because of the fuel costs for the rendezvous transfer. This equation indicates that an Aten asteroid is most likely the best target for mining. Other types, such as Apollo and Amor, need a longer transfer time and shorter mining seasons. An example of a Hohmann transfer for an Apollo asteroid is given in Fig. 3.5.

Fig. 3.5

Mission to an Apollo type of asteroid

These missions imply Hohmann transfers with: a rendezvous near but before aphelion for least changes in velocity; a short aphelion-centred mining season; and a post-aphelion departure for Earth-return, for least ∆v _r . Such missions have a longer project duration with respect to Aten missions. The use of a Hohmann ellipse transfer and short mining season imply that mission duration approximates to the orbital period of the asteroid. Transfer time is normally longer for Apollo asteroids than Aten ones. Longer project durations affect negatively the net present value. Shorter mining seasons need tougher demand specifications for the equipment for the same mass of metals returned to Earth orbit, which affects the investment costs.