Editing Space elevator (section)

==Physics==

===Apparent gravitational field===
An Earth space elevator cable rotates along with the rotation of the Earth. Therefore, the cable, and objects attached to it, would experience upward centrifugal force in the direction opposing the downward gravitational force. The higher up the cable the object is located, the less the gravitational pull of the Earth, and the stronger the upward centrifugal force due to the rotation, so that more centrifugal force opposes less gravity. The centrifugal force and the gravity are balanced at geosynchronous equatorial orbit (GEO). Above GEO, the centrifugal force is stronger than gravity, causing objects attached to the cable there to pull ''upward'' on it. Because the counterweight, above GEO, is rotating about the Earth faster than the natural orbital speed for that altitude, it exerts a centrifugal pull on the cable and thus holds the whole system aloft.

The net force for objects attached to the cable is called the ''apparent gravitational field''. The apparent gravitational field for attached objects is the (downward) gravity minus the (upward) centrifugal force. The apparent gravity experienced by an object on the cable is zero at GEO, downward below GEO, and upward above GEO.

The apparent gravitational field can be represented this way:<ref name="aravind"/>{{rp|Table 1}}

{{block indent|The downward force of actual [[Newton's law of universal gravitation|gravity]] ''decreases'' with height: [[Newton's law of universal gravitation|<math>g_r = -GM/r^2</math>]]}}
{{block indent|The upward [[centrifugal force]] due to the planet's rotation ''increases'' with height: [[Centrifugal force|<math>a = \omega^2 r</math>]]}}
{{block indent|Together, the apparent gravitational field is the sum of the two:
{{block indent|<math>g = -\frac{GM}{r^2} + \omega^2 r</math>}}}}

where
{{block indent|''g'' is the acceleration of ''apparent'' gravity, pointing down (negative) or up (positive) along the vertical cable (m s<sup>−2</sup>),}}
{{block indent|''g<sub>r</sub>'' is the gravitational acceleration due to Earth's pull, pointing down (negative)(m s<sup>−2</sup>),}}
{{block indent|''a'' is the centrifugal acceleration, pointing up (positive) along the vertical cable (m s<sup>−2</sup>),}}
{{block indent|''G'' is the [[gravitational constant]] (m<sup>3</sup> s<sup>−2</sup> kg<sup>−1</sup>)}}
{{block indent|''M'' is the mass of the Earth (kg)}}
{{block indent|''r'' is the distance from that point to Earth's center (m),}}
{{block indent|''ω'' is Earth's rotation speed (radian/s).}}

At some point up the cable, the two terms (downward gravity and upward centrifugal force) are equal and opposite. Objects fixed to the cable at that point put no weight on the cable. This altitude (r<sub>1</sub>) depends on the mass of the planet and its rotation rate. Setting actual gravity equal to centrifugal acceleration gives:<ref name="aravind"/>{{rp|p. 126}}
{{block indent|<math>r_1 = \left(\frac{GM}{\omega^2}\right)^\frac{1}{3}</math>}}

This is {{convert|35786|km|mi|0|abbr=on}} above Earth's surface, the altitude of geostationary orbit.<ref name="aravind"/>{{rp|Table 1}}

On the cable ''below'' geostationary orbit, downward gravity would be greater than the upward centrifugal force, so the apparent gravity would pull objects attached to the cable downward. Any object released from the cable below that level would initially accelerate downward along the cable. Then gradually it would deflect eastward from the cable. On the cable ''above'' the level of stationary orbit, upward centrifugal force would be greater than downward gravity, so the apparent gravity would pull objects attached to the cable ''upward''. Any object released from the cable ''above'' the geosynchronous level would initially accelerate ''upward'' along the cable. Then gradually it would deflect westward from the cable.

===Cable section===
Historically, the main technical problem has been considered the ability of the cable to hold up, with tension, the weight of itself below any given point. The greatest tension on a space elevator cable is at the point of geostationary orbit, {{convert|35786|km|mi|0|abbr=on}} above the Earth's equator. This means that the cable material, combined with its design, must be strong enough to hold up its own weight from the surface up to {{convert|35786|km|mi|0|abbr=on}}. A cable which is thicker in cross section area at that height than at the surface could better hold up its own weight over a longer length. How the cross section area tapers from the maximum at {{convert|35786|km|mi|0|abbr=on}} to the minimum at the surface is therefore an important design factor for a space elevator cable.

To maximize the usable excess strength for a given amount of cable material, the cable's cross section area would need to be designed for the most part in such a way that the [[Stress (mechanics)|stress]] (i.e., the tension per unit of cross sectional area) is constant along the length of the cable.<ref name="aravind" /><ref>Artuković, Ranko (2000). [http://www.zadar.net/space-elevator/ "The Space Elevator".] zadar.net</ref> The constant-stress criterion is a starting point in the design of the cable cross section area as it changes with altitude. Other factors considered in more detailed designs include thickening at altitudes where more space junk is present, consideration of the point stresses imposed by climbers, and the use of varied materials.<ref name="PhaseII"/> To account for these and other factors, modern detailed designs seek to achieve the largest ''[[Factor of safety#Margin of safety|safety margin]]'' possible, with as little variation over altitude and time as possible.<ref name="PhaseII"/> In simple starting-point designs, that equates to constant-stress.

For a constant-stress cable with no safety margin, the cross-section-area as a function of distance from Earth's center is given by the following equation:<ref name="aravind" />

{{CSS image crop
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|bSize = 375
|cWidth = 330
|cHeight = 135
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|Description = Several taper profiles with different material parameters
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{{block indent|<math>A( r ) = A_s \exp\left[ \frac{\rho g R^2}{T}\left( \frac{1}{R}+\frac{R^2}{2R_g^3}-\frac{1}{r}-\frac{r^2}{2R_g^3} \right) \right]</math>}}

where
{{block indent|<math>g</math> is the gravitational acceleration at Earth's surface (m·s<sup>−2</sup>),}}
{{block indent|<math>A_s</math> is the cross-section area of the cable at Earth's surface (m<sup>2</sup>),}}
{{block indent|<math>\rho</math> is the density of the material used for the cable (kg·m<sup>−3</sup>),}}
{{block indent|<math>R</math> is the Earth's equatorial radius,}}
{{block indent|<math>R_g</math> is the radius of geosynchronous orbit,}}
{{block indent|1=<math>T</math> is the stress the cross-section area can bear without [[Yield (engineering)|yielding]] (N·m<sup>−2</sup>), its elastic limit.}}

Safety margin can be accounted for by dividing T by the desired safety factor.<ref name="aravind" />

===Cable materials===
Using the above formula, the ratio between the cross-section at geostationary orbit and the cross-section at Earth's surface, known as taper ratio, can be calculated:<ref group="note">Specific substitutions used to produce the factor {{val|4.85|e=7}}:{{block indent|<math>A(R_g)/A_s = \exp \left[ \frac{\rho \times 9.81 \times (6.378\times 10^6)^2 } {T} \left( \frac{1}{6.378\times 10^6} + \frac{(6.378\times 10^6)^2}{2 (4.2164\times 10^7)^3} - \frac{1}{4.2164\times 10^7} - \frac{(4.2164\times 10^7)^2}{2 (4.2164\times 10^7)^3}\right)\right]</math>}}</ref>{{block indent|<math>A(R_g)/A_s = \exp \left[\frac{\rho}{T}\times 4.85\times 10^7\right]</math> }}
[[File:Space Elevator Taper Ratio.svg|thumb|upright=1.2|Taper ratio as a function of specific strength]]
{| class="wikitable" style="text-align:left"
|+Taper ratio for some materials<ref name="aravind" />
|-
!Material!!Tensile strength<br />(MPa)!!Density<br />(kg/m<sup>3</sup>)!![[Specific strength]]<br />(MPa)/(kg/m<sup>3</sup>)!!Taper ratio
|-
|[[Steel]] || 5,000 || 7,900 || 0.63 ||{{val|1.6|e=33}}
|-
|[[Kevlar]] || 3,600 || 1,440 || 2.5 ||{{val|2.5|e=8}}
|-
|[[UHMWPE]] @23°C || 3,600 || 0,980 || 3.7 ||{{val|5.4|e=6}}
|-
|Single wall [[carbon nanotube]] || 130,000 || 1,300 || 100 || 1.6
|}
The taper ratio becomes very large unless the specific strength of the material used approaches 48 (MPa)/(kg/m<sup>3</sup>). Low specific strength materials require very large taper ratios which equates to large (or astronomical) total mass of the cable with associated large or impossible costs.