(planet center)} $row column {geostationary radius

(planet surface)} $row column {geostationary speed

(meters/second)} $row column {Cable mass

(Kg)} $row column {Cable mass

(Empire State Buildings)} dict for {body info} $bodies { set row [$tab row] $row column $body set mass [lindex [dict get $info mass] 0] if {[dict exists $info μ]} { set const [lindex [dict get $info μ] 0] } else { set const [expr {$mass*$G}] dict set bodies $body μ $const } set T [expr {[lindex [dict get $info rotation_period] 0]*3600.0}] set R [lindex [dict get $info planet_radius] 0] set radius3 [expr {($const*$T**2)/(4.0*$pi**2)}] set radius [expr {pow($radius3,$t3)}] set length [expr {$radius-$R}] set speed [expr {sqrt($const/$radius)}] set cable_mass [expr {$length*$cable_mass_unit}] $row column [signif $radius 6] $row column [signif $length 6] $row column [signif $speed 6] $row column [format %e [signif $cable_mass 6]] $row column [signif [expr {$cable_mass/$empire}] 6] } para {Even for the shortest cable on Psyche, we are talking about a lot of mass of cable. And for my purposes I like to have something to compare it too. So, I chose the mass of [link https://en.wikipedia.org/wiki/Empire_State_Building {Empire State Building}] (331,122,430 kg).} para {Ok, so far we have established that for the Asteroid 16 Psyche, we need to lob the mass of 2.1 Empire state buildings worth of cable into an 87.5 m/s orbit. For Earth we would need 368x as much material, boosted to a much faster speed. That is what makes a space elevator for Earth utterly impractical. But for Psyche... it won't be cheap, but we may well be in the realm of what technology could pull off. In fact, this cable weighs less than some of the planetary settlements I have envisioned, and a lot less than some of the spacecraft.} set modulus [expr {200*1e9}] set length 100000 set info [dict get $bodies {16 Psyche}] set const [dict get $info μ] set speed [expr {sqrt($const/$length)}] set mass [expr {$length*$cable_mass_unit*$cable_area + $length*$cable_mass_unit*$cable_area}] set energy [expr {$speed**2*$mass}] set load [expr {$mass*0.5*$speed**2/$length}] set UL [my tag UL] $UL item "Length: [signif $length 6] m" $UL item "Speed: [signif $speed 6] m/s" $UL item "Total Mass: [format %e [signif $mass 6]] kg ( [signif [expr $mass/$empire] 3] Empire State Buildings)" $UL item "Orbital Kinetic Energy: [format %e [signif $energy 6]] kg⋅m⋅s−2" set IL [$UL tag UL] $IL item "([signif [expr {$energy/4184000000.0}] 6] Tons of TNT)" $IL item "([signif [expr {$energy/(33375000.0*168.0)}] 5] Saturn V First stages)" $UL item "Load: [format %e [signif $load 4]] N" $UL item "Stress: [format %e [signif [expr {$load/$cable_area}] 4]] Pa" $UL item "Stretch: [signif [expr {$load*$length/($cable_area*$modulus)}] 4] meters" para {Ok... so, still a bit of an engineering challenge. The Stretch factor tells you how much the cable is going to elongate under the load of the counterweight. Stress is how much load is on a unit section cable. Repeating the process for Ceres, but with a 730km cable:} set length 730000 set info [dict get $bodies {1 Ceres}] set const [dict get $info μ] set speed [expr {sqrt($const/$length)}] set mass [expr {$length*$cable_mass_unit*$cable_area + $length*$cable_mass_unit*$cable_area}] set energy [expr {$speed**2*$mass}] set load [expr {$mass*0.5*$speed**2/$length}] set UL [my tag UL] $UL item "Length: [signif $length 6] m" $UL item "Speed: [signif $speed 6] m/s" $UL item "Total Mass: [format %e [signif $mass 6]] kg ( [signif [expr $mass/$empire] 3] Empire State Buildings)" $UL item "Orbital Kinetic Energy: [format %e [signif $energy 6]] kg⋅m⋅s−2" set IL [$UL tag UL] $IL item "([signif [expr {$energy/4184000000.0}] 6] Tons of TNT)" $IL item "([signif [expr {$energy/(33375000.0*168.0)}] 5] Saturn V First stages)" $UL item "Stress: [format %e [signif [expr {$load/$cable_area}] 4]] Pa" $UL item "Stretch: [signif [expr {$load*$length/($cable_area*$modulus)}] 4] meters" para {And for completeness, let's try the Moon with a 4630km cable:} set length 4630000 set info [dict get $bodies {Luna}] set const [dict get $info μ] set speed [expr {sqrt($const/$length)}] set mass [expr {$length*$cable_mass_unit*$cable_area + $length*$cable_mass_unit*$cable_area}] set energy [expr {$speed**2*$mass}] set load [expr {$mass*0.5*$speed**2/$length}] set UL [my tag UL] $UL item "Length: [signif $length 6] m" $UL item "Speed: [signif $speed 6] m/s" $UL item "Total Mass: [format %e [signif $mass 6]] kg ( [signif [expr $mass/$empire] 3] Empire State Buildings)" $UL item "Orbital Kinetic Energy: [format %e [signif $energy 6]] kg⋅m⋅s−2" set IL [$UL tag UL] $IL item "([signif [expr {$energy/4184000000.0}] 6] Tons of TNT)" $IL item "([signif [expr {$energy/(33375000.0*168.0)}] 5] Saturn V First stages)" $UL item "Stress: [format %e [signif [expr {$load/$cable_area}] 4]] Pa" $UL item "Stretch: [signif [expr {$load*$length/($cable_area*$modulus)}] 4] meters" para {And finally... Earth. With a 35786km long cable...} set length 35786000 set info [dict get $bodies {Earth}] set const [dict get $info μ] set speed [expr {sqrt($const/$length)}] set mass [expr {$length*$cable_mass_unit*$cable_area + $length*$cable_mass_unit*$cable_area}] set energy [expr {$speed**2*$mass}] set load [expr {$mass*0.5*$speed**2/$length}] set UL [my tag UL] $UL item "Length: [signif $length 6] m" $UL item "Speed: [signif $speed 6] m/s" $UL item "Total Mass: [format %e [signif $mass 6]] kg ( [signif [expr $mass/$empire] 3] Empire State Buildings)" $UL item "Orbital Kinetic Energy: [format %e [signif $energy 6]] kg⋅m⋅s−2" set IL [$UL tag UL] $IL item "([signif [expr {$energy/4184000000.0}] 6] Tons of TNT)" $IL item "([signif [expr {$energy/(33375000.0*168.0)}] 5] Saturn V First stages)" $UL item "Stress: [format %e [signif [expr {$load/$cable_area}] 4]] Pa" $UL item "Stretch: [signif [expr {$load*$length/($cable_area*$modulus)}] 4] meters" para {Assuming this is science fiction and we have [link https://en.wikipedia.org/wiki/Fusion_rocket {fusion powered rocket engines}], and [link https://en.wikipedia.org/wiki/Mass_driver {Mass Drivers}] at our disposal, the energy requirements to get the material in orbit are not all that crazy.} para {But we have more to wrestle with than energy to orbit. Note that stress column. I picked the cable on the Golden Gate Bridge because we know [link https://www.goldengate.org/bridge/history-research/statistics-data/design-construction-stats/ {roughly how much weight it can hold}]. The strength of wire rope is a [link {https://www.nap.edu/read/23338/chapter/8#151} {very well understood}] problem in Engineering. The bridge is still standing after nearly a century, so they must have gotten the answer right. Long story short, the strength of a cable increases with the cross sectional area. And you can build up immense cross sectional areas by adding together smaller wires.} my tag img src /[my request get PREFIX_URI]/susp1.gif my tag img width 400 src /[my request get PREFIX_URI]/golden-gate-bridge-cable-cross-section.jpg set load [expr {(21772000*0.5)*9.8}] para {Two cables on the Golden Gate Bridge hold up about 21,772,000 kg of structure. So we can say that each cable is holding up half. But we are concerned with WEIGHT so, we have to remember to multiply mass by Gravity (9.8 m/s^2) to get Newtons: [format %e [signif $load 5]] N. The stress on those cables is the load / area. Thus we can guestimate a safe stress to be [set GGB_strain [format %e [signif [expr {$load/$cable_area}] 5]]] [link https://en.wikipedia.org/wiki/Pascal_(unit) Pascals].} para {However raw strength isn't the complete answer. Steel rope (and indeed virtually every structural material) [link {https://www.assemblyspecialty.com/guide-to-wire-rope/technical-information/physical-properties-of-wire-rope/} {stretches under load}]. Below the yield strength, steel stretches by a relationship between stress and strain known as the [link https://www.engineeringtoolbox.com/young-modulus-d_417.html {Young's Modulus}]. Stretch the material to [link {https://www.engineeringtoolbox.com/young-modulus-d_417.html#yield_strength} {the yield strength}]. and you will permanently deform the material. The material will continue stretching until the [link https://www.engineeringtoolbox.com/young-modulus-d_417.html#ultimate_tensile_strength {ultimate tensile strength}], at which point it will break. For generic structural steel, the Young's Modulus is 2.00e11, the Yield Strength is 2.5e8 Pa, and the Ultimate Tensile Strength is 4.0e+08 Pa.} para {Our calculation for cable strain on the Golden Gate Bridge ($GGB_strain Pa) is only a fraction of the yield strength of steel (2.5e8 Pa). This is by design. A bridge needs to handle a lot of strain in addition to its own weight. In fact the operators of the Golden Gate bridge never allow the strain to exceed 40% of the Tensile strength on the cables.} para {For our Psyche cable we have a stress of 1.152000e+08 Pa. An accountant might say we have [signif [expr {(1.0-1.152000e+08/1.604200e+08)*100}] 3]% more rope than would be strictly called for if this cable were on a bridge. What happens if we make the cable 28% smaller?} set length 100000 set scale [expr {1.152000e+08/1.604200e+08}] set cable_area [signif [expr {(.92*0.5)**2*$pi*$scale}] 4] set info [dict get $bodies {16 Psyche}] set const [dict get $info μ] set speed [expr {sqrt($const/$length)}] set mass [expr {$length*$cable_mass_unit*$cable_area + $length*$cable_mass_unit*$cable_area}] set energy [expr {$speed**2*$mass}] set load [expr {$mass*0.5*$speed**2/$length}] set UL [my tag UL] $UL item "Speed: [signif $speed 6] m/s" $UL item "Cable Area: [signif $cable_area 4] m^2" $UL item "Total Mass: [format %e [signif $mass 6]] kg ([signif [expr {$mass/$empire}] 3] empire state buildings)" $UL item "Kinetic Energy: [format %e [signif $energy 6]] kg⋅m⋅s−2 ([signif [expr {$energy/4184000000.0}] 6] Tons of TNT) ([signif [expr {$energy/(33375000.0*168.0)}] 5] Saturn V First stages)" $UL item "Load: [format %e [signif $load 4]] N" $UL item "Stress: [format %e [signif [expr {$load/$cable_area}] 4]] Pa" $UL item "Stretch: [signif [expr {$load*$length/($cable_area*$modulus)}] 4] meters" para {As the size of the cable goes down, so does its mass, and so too does the load needed to keep that mass aloft. The strain remains constant.} para {If a 28% reduction has such a profound change, what about 98%?} set cable_area [signif [expr {(.92*0.5)**2*$pi*0.02}] 4] set length 100000 set scale [expr {.02}] set info [dict get $bodies {16 Psyche}] set const [dict get $info μ] set speed [expr {sqrt($const/$length)}] set mass [expr {$length*$cable_mass_unit*$cable_area + $length*$cable_mass_unit*$cable_area}] set energy [expr {$speed**2*$mass}] set load [expr {$mass*0.5*$speed**2/$length}] set UL [my tag UL] $UL item "Speed: [signif $speed 6] m/s $scale" $UL item "Cable Area: [signif $cable_area 4] m^2" $UL item "Total Mass: [format %e [signif $mass 6]] kg ([signif [expr {$mass/$empire}] 3] empire state buildings)" $UL item "Kinetic Energy: [format %e [signif $energy 6]] kg⋅m⋅s−2 ([signif [expr {$energy/4184000000.0}] 6] Tons of TNT) ([signif [expr {$energy/(33375000.0*168.0)}] 5] Saturn V First stages)" $UL item "Load: [format %e [signif $load 4]] N" $UL item "Stress: [format %e [signif [expr {$load/$cable_area}] 4]] Pa" $UL item "Stretch: [signif [expr {$load*$length/($cable_area*$modulus)}] 4] meters" para {Our case for space elevator on Psyche is more or less made. Decide on the mass of your orbiting station, and pretty much size your cable accordingly. The smaller the station, the less cable you need. The less cable you need... the less cable you need. Most of the load is dealing with the weight of the cable.} set cable_area [signif [expr {(0.05*0.5)**2*$pi}] 4] para {You will note that for our larger bodies, though, the stress on the rope exceeds not just the Yield strength, but the ultimate tensile strength of Steel. But what if we learned from our little experiment using something with a higher tensile strength? I've picked out a cable that can be purchased, today, [link http://www.hampidjan.com.au/dynice-warp-winch-ropes.html {from a catalogue}]. It's 50mm in diameter. It has a breaking strength of [format %e [expr {169*1000*9.8/$cable_area}]] Pa. And a mass of 1.3 kg/m. Let's try that out on Ceres, the Moon, and Earth.} my tag img width 500 src /[my request get PREFIX_URI]/dynice-warp-winch-diagram.jpg set cable_mass_unit 1.3 para "Ceres:" set length 730000 set scale [expr {6.147000e+08/5.667900e+09}] set info [dict get $bodies {1 Ceres}] set const [dict get $info μ] set speed [expr {sqrt($const/$length)}] set mass [expr {$length*$cable_mass_unit + $length*$cable_mass_unit}] set energy [expr {$speed**2*$mass}] set load [expr {$mass*0.5*$speed**2/$length}] set UL [my tag UL] $UL item "Speed: [signif $speed 6] m/s" $UL item "Total Mass: [format %e [signif $mass 6]] kg ([signif [expr {$mass/$empire}] 3] empire state buildings)" $UL item "Kinetic Energy: [format %e [signif $energy 3]] kg⋅m⋅s−2" $UL item "Load: [format %e [signif $load 4]] N" $UL item "Stress: [format %e [signif [expr {$load/$cable_area}] 4]] Pa" para "Luna:" set length 4630000 set info [dict get $bodies {Luna}] set const [dict get $info μ] set speed [expr {sqrt($const/$length)}] set mass [expr {$length*$cable_mass_unit + $length*$cable_mass_unit}] set energy [expr {$speed**2*$mass}] set load [expr {$mass*0.5*$speed**2/$length}] set UL [my tag UL] $UL item "Speed: [signif $speed 6] m/s" $UL item "Total Mass: [format %e [signif $mass 6]] kg ([signif [expr {$mass/$empire}] 3] empire state buildings)" $UL item "Kinetic Energy: [format %e [signif $energy 6]] kg⋅m⋅s−2" $UL item "Stress: [format %e [signif [expr {$load/$cable_area}] 4]] Pa" para {Earth:} set length 35786000 set scale 0.0203 set info [dict get $bodies {Earth}] set const [dict get $info μ] set speed [expr {sqrt($const/$length)}] set mass [expr {$length*$cable_mass_unit + $length*$cable_mass_unit}] set energy [expr {$speed**2*$mass}] set load [expr {$mass*0.5*$speed**2/$length}] set UL [my tag UL] $UL item "Speed: [signif $speed 6] m/s" $UL item "Total Mass: [format %e [signif $mass 6]] kg ([signif [expr {$mass/$empire}] 3] empire state buildings)" $UL item "Kinetic Energy: [format %e [signif $energy 6]] kg⋅m⋅s−2" $UL item "Stress: [format %e [signif [expr {$load/$cable_area}] 4]] Pa" para {Earth... we just can't seem to make it work for Earth. Luna is inside the breaking strength of the cable, but not within any kind of comfortable safety margin. For Ceres, the 50mm cable is well inside the breaking strength.} para {So in the end, how practical would a space elevator be?} my tag h3 {Psyche} para {The stress and strain needed for a cable would permit massive loads to be moved into orbit using conventional steel. That cable system could scale to practically any size station in orbit.} my tag h3 {Ceres} para {A practical cargo/passenger system could be devised using advanced composite cables. But you aren't moving the likes of ships into orbit with it.} my tag h3 {Luna} para {A cable is possible as a tourist attraction or for moving high-value/low volume cargo. But it's not something that would be cost effective to operate, as the cable will likely fatigue quickly under normal working loads barring it being made of an exotic material.} my tag h3 {Earth} para {Not happening. End of statement.} para {Images credits:} set ul [my tag ul] foreach url { https://wikimedia.org/api/rest_v1/media/math/render/svg/1385eed5863f0e40f86905c95cd3d4af7788ce9e https://wikimedia.org/api/rest_v1/media/math/render/svg/07005f1d6ce59ad5df80225ee26642603650be0e https://wikimedia.org/api/rest_v1/media/math/render/svg/c597ae5584d3dd6749cf0c49908afa9df45e9c5a https://www.scienceabc.com/wp-content/uploads/2018/08/centripital-force.jpg https://www.engineersedge.com/physics/centrifugal-force.gif https://wikimedia.org/api/rest_v1/media/math/render/svg/c86493d99fd1b940ca6e1ed1fdc0157c86580263 https://technologystudent.com/images6/susp1.gif https://c8.alamy.com/comp/F7278F/multi-colored-strands-of-a-bridge-suspension-cable-from-the-new-span-F7278F.jpg https://www.inside-guide-to-san-francisco-tourism.com/images/golden-gate-bridge-cable-cross-section.jpg http://www.hampidjan.com.au/images/dynice-warp-winch-diagram.jpg } { $ul item $url }