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| . Habitat wheel, 2000 kg/m², W = 200 meters wide, R = 140 meters radius. Mass = 2 π R W dens = 3.5e8 kg . Assume 1 gee, 9.8 m/s² = ω² R = ( 2 π freq )² R, so ω = √0.07 = 0.26458 rad/s, so freq = 0.042 Hz = 2.52 rpm . Edge velocity V = ω R = 37 m/s . Angular momentum L = V R M = 1.81e12 kg m²/s |
== Bad example, not enough asteroid angular momentum == |
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| . Shield 20 m thick, density 1000 kg/m³ 150 meter inner radius, 170 meter outer radius, 220 meter inner width, 240 meter outer width. . Shield Volume = πR₁²W₁ - πR₀²W₀ = π ( 170^2*240 - 150^2*220 ) m³ = 1986000 m³, Mass ≈ 2e9 kg . Volume of round asteroid = 1986000 m³ = 4 π R³ / 3 so R ≈ 78 meters . Surface gravity of round asteroid = G M / R² = 6.67408e-11 m³/ kg s² × 2e9 kg / 78² m² = 2.2e-5 m/s² . Maximum asteroid rotation rate ω = √ g/R = √ ( 2.2e-5 m/s² / 78 m ) = 5.3e-4 rad/s 3.3 . Maximum asteroid angular momentum L = 0.4 M R² ω = 0.4 × 2e9 × 78² × 5.3e-4 = 2e11 kg m²/s |
. Habitat wheel, 1000 kg/m², W = 200 meters wide, R = 140 meters radius. Mass = 2 π R W dens = 1.8e8 kg . Assume 1 gee, 9.8 m/s² = ω² R = ( 2 π freq )² R, so ω = √0.07 = 0.26458 rad/s, so freq = 0.042 Hz = 2.53 rpm . Habitat Edge velocity V = ω R = 37 m/s . Angular momentum L = V R M = 9.1e11 kg m²/s |
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| We need a factor of 9 more ... hence the shielding must be at least 9 times heavier and thicker (1.8e10 kg), and the source asteroid larger than 160 meters diameter. | . Assume Shield thickness 20 m thick, density 1200 kg/m³ . 150 meter inner radius, 170 meter outer radius, 220 meter inner width, 260 meter outer width . Shield Volume = πR₁²W₁ - πR₀²W₀ = π ( 170^2*260 - 150^2*220 ) m³ = 2360000 m³, Mass ≈ 9.7e9 kg . Volume of round asteroid = 1986000 m³ = 4 π R³ / 3 so R ≈ 124 meters . Surface gravity of round asteroid = G M / R² = 6.67408e-11 m³/ kg s² × 9.7e9 kg / 124² m² = 4.2e-5 m/s² . Assume maximum centrifugal acceleration 0.8 × gravity . thus ω < √( 0.8 × 4/3 × G × ρ ) = 5.18E-4 rad/s → 3.37 hours period . Maximum asteroid rotation rate ω = √(g/R) = √( 0.8 * 4.2e-5 m/s² / 78 m ) = 5.18e-4 rad/s . Maximum asteroid angular momentum L = 0.4 M R V = 0.4 × 2e9 × 78² × 5.3e-4 = 3.1e10 kg m²/s |
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| Near Earth asteroid 101955 Bennu has a mass of 7.3e10 kg, a mean density of 1200 kg/m³, a mean radius of 245 m, and a rotation period of 15500 seconds. L = 0.4 M R² ω = 7e11 kg m²/s | We need a factor of 30 more ... hence the shielding must be at least 8 times heavier and thicker (1.8e10 kg), and the source asteroid larger than 245 meters radius and spinning fast. [[ attachment:astrohabitat.ods | Libreoffice Calc ods spreadsheet ]] . . . . (download free, open source libreoffice [[ https://www.libreoffice.org/download/download/ | here ]] ) Near Earth asteroid 101955 Bennu has a mass of 7.3e10 kg, a mean density of 1200 kg/m³, a mean radius of 245 m, and a rotation period of 15500 seconds. L = 0.4 M R² ω = 7e11 kg m²/s, not quite enough! |
Large Asteroid Habitat
Bad example, not enough asteroid angular momentum
- Habitat wheel, 1000 kg/m², W = 200 meters wide, R = 140 meters radius. Mass = 2 π R W dens = 1.8e8 kg
- Assume 1 gee, 9.8 m/s² = ω² R = ( 2 π freq )² R, so ω = √0.07 = 0.26458 rad/s, so freq = 0.042 Hz = 2.53 rpm
- Habitat Edge velocity V = ω R = 37 m/s
- Angular momentum L = V R M = 9.1e11 kg m²/s
- Assume Shield thickness 20 m thick, density 1200 kg/m³
- 150 meter inner radius, 170 meter outer radius, 220 meter inner width, 260 meter outer width
Shield Volume = πR₁²W₁ - πR₀²W₀ = π ( 1702*260 - 1502*220 ) m³ = 2360000 m³, Mass ≈ 9.7e9 kg
- Volume of round asteroid = 1986000 m³ = 4 π R³ / 3 so R ≈ 124 meters
- Surface gravity of round asteroid = G M / R² = 6.67408e-11 m³/ kg s² × 9.7e9 kg / 124² m² = 4.2e-5 m/s²
- Assume maximum centrifugal acceleration 0.8 × gravity
thus ω < √( 0.8 × 4/3 × G × ρ ) = 5.18E-4 rad/s → 3.37 hours period
- Maximum asteroid rotation rate ω = √(g/R) = √( 0.8 * 4.2e-5 m/s² / 78 m ) = 5.18e-4 rad/s
- Maximum asteroid angular momentum L = 0.4 M R V = 0.4 × 2e9 × 78² × 5.3e-4 = 3.1e10 kg m²/s
We need a factor of 30 more ... hence the shielding must be at least 8 times heavier and thicker (1.8e10 kg), and the source asteroid larger than 245 meters radius and spinning fast.
Libreoffice Calc ods spreadsheet . . . . (download free, open source libreoffice here )
Near Earth asteroid 101955 Bennu has a mass of 7.3e10 kg, a mean density of 1200 kg/m³, a mean radius of 245 m, and a rotation period of 15500 seconds. L = 0.4 M R² ω = 7e11 kg m²/s, not quite enough!