Differences between revisions 1 and 17 (spanning 16 versions)
 ⇤ ← Revision 1 as of 2019-12-10 19:29:19 → Size: 1353 Editor: KeithLofstrom Comment: ← Revision 17 as of 2019-12-11 00:34:48 → ⇥ Size: 2620 Editor: KeithLofstrom Comment: Deletions are marked like this. Additions are marked like this. Line 2: Line 2: -----The habitat geometry is a large cylinder, illuminated by diffusing mirrors on the axis from an external focusing mirror. The habitat is surrounded by a thick shield of dustball asteroid material. Line 3: Line 5: . 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 Dustball asteroid rotation rates are quite limited by centrifugal disassembly. For 1200 kg/m³ and a 1.25x safety factor, the rotation period must be longer than 3.37 hours or equatorial material will spin away. The dustball asteroid 101955 Bennu has a rotation period of 4.3 hours, close to the limit, and most asteroids rotate more slowly. Asteroids do not spin too fast; if you hope to spin up a habitat with the angular momentum in the shielding mass, they spin too slowly. Line 8: Line 7: . 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 || {{ attachment:homewheel.png | | width=600 }} || {{ attachment:homewheelmirror.png | | width=180 }} || Line 15: Line 9: 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. == Bad example, not enough asteroid angular momentum == Line 17: Line 11: 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 . 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 ( 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 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!MoreLater

# Large Asteroid Habitat

The habitat geometry is a large cylinder, illuminated by diffusing mirrors on the axis from an external focusing mirror. The habitat is surrounded by a thick shield of dustball asteroid material.

Dustball asteroid rotation rates are quite limited by centrifugal disassembly. For 1200 kg/m³ and a 1.25x safety factor, the rotation period must be longer than 3.37 hours or equatorial material will spin away. The dustball asteroid 101955 Bennu has a rotation period of 4.3 hours, close to the limit, and most asteroids rotate more slowly. Asteroids do not spin too fast; if you hope to spin up a habitat with the angular momentum in the shielding mass, they spin too slowly.

## 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!

AsteroidHabitat (last edited 2020-08-20 16:13:15 by KeithLofstrom)