Jenab6
Ceres is a spherical asteroid in the main asteroid belt. It's mass and radius and rotation period are:
M = 9.43e20 kilograms
R = 487300 meters
Pr = 32667.012 seconds
Someday, there might be people living on Ceres, and when there are there will undoubtedly be some who demand that other people give them money, or else be robbed, beaten, imprisoned, exiled, or killed, and, at the same time, presenting good-sounding reasons for their extortions. Also, there will be patriots on Ceres who are too smart to be deceived by these sophistries, who will be seeking to whack the tax-collectors and the bureaucrats and officials who fatten themselves upon other people's labor.
On Ceres, assassination can become a sport, a supreme challenge of marksmanship. Players will need a .357 magnum pistol and some bullets.
Ceres' gravitational parameter, GM = 6.2938e10 m³ sec⁻²
Ceres' circular orbit speed, 1.5 meters out from the equator, is
Vo = sqrt [ GM / (R+1.5) ] = 359.383 m/s
Ceres' escape speed, from a point 1.5 meters out from the equator, is
Ve = sqrt [ 2GM / (R+1.5) ] = 508.244 m/s
Ceres' tangential rotational speed at 1.5 meters out from the equator is
Vr = 2πR / Pr = 93.728 m/s
The muzzle speed of a .357 magnum bullet is 430 m/s. Notice that if the patriot fires a bullet in the equatorial plane in the direction of his tangent velocity, the bullet will be going 523.728 m/s relative to the center of Ceres, which is faster than Ceres' escape speed. On the other hand, if he fires a bullet in the opposite direction, the bullet will be moving 336.272 m/s, which is less than Ceres' circular orbit speed.
So the patriot shoots the bullet in the direction parallel with the north rotational axis of Ceres. What happens?
Cerescentric state vector for bullet.
x = +487301.5 meters
y = 0
z = 0
Vx = 0
Vy = +93.728 m/s
Vz = +430 m/s
Bullet's Cerescentric distance and speed.
r = sqrt( x² + y² + z² ) = 487301.5 meters
V = sqrt[(Vx)² + (Vy)² + (Vz)²] = 440.096 m/s
Semimajor axis of bullet's orbit.
a = 1 / { 2/r − V²/(GM) } = 973837.021 meters
Period of bullet's orbit.
P = 2π sqrt[ a³ / (GM) ] = 24068.622 seconds
Angular momentum per unit mass in bullet's orbit.
hx = y Vz − z Vy = 0
hy = z Vx − x Vz = −209539645.0 m² sec⁻¹
hz = x Vy − y Vx = +45673674.5 m² sec⁻¹
h = sqrt[ (hx)² + (hy)² + (hz)² ] = 214459663.7 m² sec⁻¹
Eccentricity of bullet's orbit.
e = sqrt[ 1 − h² / (GMa) ] = 0.499607
Inclination of bullet's orbit, relative to Ceres' equator.
i = ArcCos( hz / h ) = 77.7035°
The bullet goes into an elliptical orbit around Ceres, with a period of 6h 41m 8.622 s, an eccentricity of about 0.5, an inclination of about 77.7 degrees, and a pericerian altitude of 1.5 meters. Each time the bullet crosses Ceres equator at pericerian, its Cerian longitude is 265.243° antispinward (west), or 94.757° spinward (east), of where it crossed the previous time.
So the patriot's job is to find out where that nasty taxman will be at some future moment, then go to a position 94.757° west of there, carefully mount his pistol on some sort of immovable stand so that it is pointing north, wait until exactly 24068.622 seconds before the bureaucrat in question is due to arrive, then pull the trigger. The pistol will fire silently, since there's no air to transmit sound, and the bullet will fly away on its errand of justice.
Now, of course, this is a sport. The timing and aiming involved, not to mention the uncertainties in the manufacture of the bullets, will make most of these shots miss. The patriot is, after all, trying to shoot somebody who is on the other end of a 5717-kilometer trajectory. He might need several tries. But imagine the psychological effect on the taxman when he sees chips of rock suddenly fly from the side of his office building. The nice thing about it is that there is practically no risk of being caught by the local police, since they can't hear the gunshot, and if one location is being watched, then another will do if a suitable change of trajectory is made.
If by some incredible good fortune, the patriot actually kills the tax-collector, the people of Ceres would rejoice, except for the remaining unacceptably presumptuous and naughty bureaucrats, who would probably be wondering which of them was to be next.
Jerry Abbott
M = 9.43e20 kilograms
R = 487300 meters
Pr = 32667.012 seconds
Someday, there might be people living on Ceres, and when there are there will undoubtedly be some who demand that other people give them money, or else be robbed, beaten, imprisoned, exiled, or killed, and, at the same time, presenting good-sounding reasons for their extortions. Also, there will be patriots on Ceres who are too smart to be deceived by these sophistries, who will be seeking to whack the tax-collectors and the bureaucrats and officials who fatten themselves upon other people's labor.
On Ceres, assassination can become a sport, a supreme challenge of marksmanship. Players will need a .357 magnum pistol and some bullets.
Ceres' gravitational parameter, GM = 6.2938e10 m³ sec⁻²
Ceres' circular orbit speed, 1.5 meters out from the equator, is
Vo = sqrt [ GM / (R+1.5) ] = 359.383 m/s
Ceres' escape speed, from a point 1.5 meters out from the equator, is
Ve = sqrt [ 2GM / (R+1.5) ] = 508.244 m/s
Ceres' tangential rotational speed at 1.5 meters out from the equator is
Vr = 2πR / Pr = 93.728 m/s
The muzzle speed of a .357 magnum bullet is 430 m/s. Notice that if the patriot fires a bullet in the equatorial plane in the direction of his tangent velocity, the bullet will be going 523.728 m/s relative to the center of Ceres, which is faster than Ceres' escape speed. On the other hand, if he fires a bullet in the opposite direction, the bullet will be moving 336.272 m/s, which is less than Ceres' circular orbit speed.
So the patriot shoots the bullet in the direction parallel with the north rotational axis of Ceres. What happens?
Cerescentric state vector for bullet.
x = +487301.5 meters
y = 0
z = 0
Vx = 0
Vy = +93.728 m/s
Vz = +430 m/s
Bullet's Cerescentric distance and speed.
r = sqrt( x² + y² + z² ) = 487301.5 meters
V = sqrt[(Vx)² + (Vy)² + (Vz)²] = 440.096 m/s
Semimajor axis of bullet's orbit.
a = 1 / { 2/r − V²/(GM) } = 973837.021 meters
Period of bullet's orbit.
P = 2π sqrt[ a³ / (GM) ] = 24068.622 seconds
Angular momentum per unit mass in bullet's orbit.
hx = y Vz − z Vy = 0
hy = z Vx − x Vz = −209539645.0 m² sec⁻¹
hz = x Vy − y Vx = +45673674.5 m² sec⁻¹
h = sqrt[ (hx)² + (hy)² + (hz)² ] = 214459663.7 m² sec⁻¹
Eccentricity of bullet's orbit.
e = sqrt[ 1 − h² / (GMa) ] = 0.499607
Inclination of bullet's orbit, relative to Ceres' equator.
i = ArcCos( hz / h ) = 77.7035°
The bullet goes into an elliptical orbit around Ceres, with a period of 6h 41m 8.622 s, an eccentricity of about 0.5, an inclination of about 77.7 degrees, and a pericerian altitude of 1.5 meters. Each time the bullet crosses Ceres equator at pericerian, its Cerian longitude is 265.243° antispinward (west), or 94.757° spinward (east), of where it crossed the previous time.
So the patriot's job is to find out where that nasty taxman will be at some future moment, then go to a position 94.757° west of there, carefully mount his pistol on some sort of immovable stand so that it is pointing north, wait until exactly 24068.622 seconds before the bureaucrat in question is due to arrive, then pull the trigger. The pistol will fire silently, since there's no air to transmit sound, and the bullet will fly away on its errand of justice.
Now, of course, this is a sport. The timing and aiming involved, not to mention the uncertainties in the manufacture of the bullets, will make most of these shots miss. The patriot is, after all, trying to shoot somebody who is on the other end of a 5717-kilometer trajectory. He might need several tries. But imagine the psychological effect on the taxman when he sees chips of rock suddenly fly from the side of his office building. The nice thing about it is that there is practically no risk of being caught by the local police, since they can't hear the gunshot, and if one location is being watched, then another will do if a suitable change of trajectory is made.
If by some incredible good fortune, the patriot actually kills the tax-collector, the people of Ceres would rejoice, except for the remaining unacceptably presumptuous and naughty bureaucrats, who would probably be wondering which of them was to be next.
Jerry Abbott