Space Elevator is Closer than we think

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online slides version

0. Introduction

Humanity either spreads across the universe or perishes completely. No other choice

–Liu.CiXin(H.G.Wells)

1. Background

General Background to different methods entering space

1.1 Rocket

Starship

1.3 Space Airplane

Skylon

1.4 Mass Driver

Spin Launch

1.2 Space Elevator

Obayashi inc

ISEC

2. Space Elevator Design from NASA(2000)

1. Deploy a minimal cable $1\mu m\times (5\text{cm}\sim 11.5\text{cm})\times 91000\text{km}$

   it can support $1238 \text{kg}$ weight

2. Increase this minimal cable to a useful capability

   each climber completes its ascent the cable would be $1.5$ stronger.

   After $207$ climbers($2.3$ years), the cable would supporting $20,000\text{kg}$ climber with a $13,000\text{kg}$ payload.

   (Additional cables of comparable capacity could be produced every $170$ days using the first cable)

   In $2.8$ years the capacity of any individual($20,000\text{kg}$) cable could be built up to ($1000,000\text{kg}$) orbiter. Payloads as large as the shuttle orbiter can be sent to the earth orbit.

3. Utilize the cable for accessing space

2.1 Cable

Carbon NanoTube

Tensile strength

	CNT	Steel	Kevlar
tensile strength($\text{GPa}$)	$130$	$<5$	3.6
density($\text{kg}/\text{m}^3$)	$1300$	$7900$	$1440$
taper ratio	$1.5$	$1.7\times 10^{33}$	$2.6\times{10^8}$

2.1.1 different cross-section area in different altitude

2.1.2 micro-scale design

epoxy-CNT compound(60%CNT-40%epoxy)

another version of cable design

2.1.3 CNT Production

1. short CNT

2. no defects cable are allowed

3. production time for each cable must be no more than one year and 100 in parallel

2.2 Initial Cable Deployer

1. $168,000\text{kg}$ spacecraft ($19,800\text{kg}$ cable) should be sent to GEO(ISS~$420,000\text{kg}$, TSS~$180,000\text{kg}$)
2. no extra power is required to deploy the cable.

1. craft is to impart a small amount of angular moments to the cable as it is initially deployed
2. transmit beacon signal, so the end of the cable can be found and retrieved on earth.

2.3 Climber

1. total mass: $619\text{kg}$($288\text{kg}$ cable)
2. the cable deployed by first climber will be shorter and stronger($91,000\text{km} < 117,00\text{km}$, $9.7\text{kg}$ > $8\text{kg}$)
3. first $207$ climbers widen the edge of the initial ribbon to $30\text{cm}$ at $200\text{km/h}$(reduce catastrophic and meteor damage). Once it reaches $30\text{cm}$ width, then thicken the ribbon.
4. expandable design as it become larger each climb
5. stuck: low altitude-> retrieve, high altitude->release
6. if power-kilogram ratio > $40$($70\text{kW}$ for $113\text{W/kg}$). strengthen cable to $20,000\text{kg}$ in $1.7$ years, $1000,000\text{kg}$ in $3.7$years. And also reduce cable damage(could reach $47.84 \text{kw/kg}$ in $2022$, $3000\text{kW}$ $62.7\text{kg}$)

2.3 Power

	Laser	Microwave
operating wavelength	$0.84\mu m$	$3.2 \text{mm}(94\text{GHz})$
transmitter system	free-electron laser + deformable mirro	phased array
transmitter area	$12\text{m}$ diametere	1km diameter
receiver system	tunned solar cells	rectennas
overall efficiency	$2$	$0.05$

2.4 Deployment

• MMH : monomethylhydrazine
• NTO : nitrogen tetroxide
• SC : space craft

2.5 Anchor

2.6 Destination

2.6 Challenges

• Lightning

  lightning-free zone

  

• Meteors&Space Debirs&Low-Earch-Orbit objects

  ribbon-design

  

  

  

• Wind

  wind-free zone and special design

  

• Atomic Oxygen

  probably the most tricky challenge

  coat with $Ni+SiO_2$ as thin as $0.16\mu m$ or $Pt$ and $Au$

  

• Electromagnetic fields

  the heating could be quickly radiate into the space

• Radiation

  more than $1000$ years in the expected environment

• Oscillation

  characteristic period $7.1$ hours.

  climber no more than $10,000\text{km/h}$

• environment impact

2.6 Advantage

• 99% reduction in cost in entering space
• less space debris
• repair and removal spacecrafts
• large-scale commercial manufacturing in microgavity space
• large-orbit solar collectors for power generation
• …

3. Modern Space Elevator Design

3.1 Cable

Graphene

similar mechanics performance as CNT, but much easier to produce in large scale

LEED: Low Energy Electron Diffraction

3.2 Climber

3.3 Power

beyond atomsphere

4. Appendix

4.1 Terminology

• LEO: Lowest Earth Orbit

  where the period of 128min or less

  $$r_{LEO} \approx 200km+R_{earth}$$

• GEO: Geostationary Equatorial Obit

  only in equator and following the direction of Earth’s rotation.

  $$\begin{aligned} r_{GEO} &= \sqrt[3]{\frac{GMT^2}{4\pi^2}}\\ &\approx 42164km\\ &\approx 25786km + R_{earth} \end{aligned}$$

4.2 Rocket

Rocket	GTO(GEO)
Falcon Heavy	$26,700\text{kg}$
Falcon 9	$8,300\text{kg}$

4.3 Space Elevator Physics

4.3.1 space elevator height and tape ratio

Notation

• $r_0$ : earth radius
• $g_0$ : surface gravity
• $r_s$ : synchronous orbit radius
• $\rho$ : density of the cable
• $A$ : cross-sectional area
• $A_s$ : cross-sectional area at synchronous orbit radius
• $r_t$ : the end of the cable
• $\sigma$ : uniform stress
• $h$ : characteristic height $h = \frac{\sigma}{\rho g_0}$

The total gravity force should equal to the centrifugal force

$$\begin{aligned} F&=\int_{r_0}^{r_t}\rho A g_0 r_0^2(1/r^2-r/r_s^3)dr = 0 \\ \Rightarrow r_t &\approx 150,000km \end{aligned}$$

assuming the stress in the cable distributed uniformly.

$$\begin{aligned} \sigma dA &= \rho g_0 r_0^2 (1/r^2-r/r_s^3)A(r)dr \\\Rightarrow A(r) &= A_s e^{3r_0^2/2hr_s}e^{(-r_0/h)(r_0)} \\\Rightarrow \frac{A_s}{A_0} &= e^{0.776r_0/h} \end{aligned}$$

4.3.2 Deploy speed

Notation

• $T_g$ : gravitational torque
• $R$ : orbit radius
• $\mu$ : earth gravity constant ($3.9\times 10^5km^3/s^2$)
• $\theta$ : maximum deviation of Z axis from the vertical
• $\omega$ : angular velocity of the earth ($7.3\times 10^{-5}s^{-1}$)
• $r$ : length of deployed cable
• $I$ : angular momentum of the spacecraft ($mr^2$)
• $m$ : weight of the cable end

$$T_g = \frac{3\mu}{R^3}|I_z - I_y|\theta$$

$$\begin{aligned} &T_g = \frac{d(I\omega)}{dt} = \omega mr\frac{dr}{dt} \\&\Rightarrow \frac{3\mu}{R^3} mr^2\theta = \omega mr\frac{dr}{dt} \\&\Rightarrow r = e^{\frac{3\mu\theta}{\omega R}t} \end{aligned}$$