186 lines
4.1 KiB
Plaintext
186 lines
4.1 KiB
Plaintext
[ From max@west.darkside.com (Erik Max Francis)]
|
||
|
||
RELATIVISTIC EFFECTS UNDER CONSTANT THRUST
|
||
|
||
A body of mass m0 is accelerating under constant thrust F. Initial
|
||
(rest) acceleration is a0 = F/m0. t is the elapsed time as measured
|
||
from the rest frame.
|
||
|
||
Effective acceleration (from the rest frame):
|
||
|
||
a = a0 (1 - v^2/c^2)^(3/2).
|
||
|
||
Accumulated velocity (from the rest frame):
|
||
|
||
v = [t^2/(a0^2 + c^2 t^2)].
|
||
|
||
Accumulated displacement (from the rest frame):
|
||
|
||
r = (a0^2 + c^2 t^2)^(1/2) - c^2/a0.
|
||
|
||
Accumulated gamma term:
|
||
|
||
gamma = (c^2 + a0^2 t^2)/c^2.
|
||
|
||
Elapsed ship time (as a function of rest time):
|
||
|
||
tau = (c/a0) ln [(1 + a0^2 t^2/c^2)^(1/2) + a0 t/c].
|
||
|
||
|
||
FUEL CONSUMPTION
|
||
|
||
A ship with mass m0 and initial mass of fuel f0 has a drive which is
|
||
capable of constant fuel consumption K and an exhaust velocity
|
||
(specific impulse) E.
|
||
|
||
Actual acceleration as a function of time:
|
||
|
||
a = E K/(m0 + f0 - K t).
|
||
|
||
Time taken to exhaust fuel:
|
||
|
||
T = f0/K.
|
||
|
||
Accumulated velocity as a function of time:
|
||
|
||
v = E [ln (m0 + f0) - ln (m0 + f0 - K t)].
|
||
|
||
Final change in velocity after exhausting fuel:
|
||
|
||
v|T = E [ln (m0 + f0) - ln m0].
|
||
|
||
Final displacement after exhausting fuel:
|
||
|
||
r|T = (E/K) [m0 [ln m0 - ln (m0 + f0)] + f0].
|
||
|
||
|
||
RELATIVISTIC FUEL CONSUMPTION
|
||
|
||
A ship with mass m0 and initial mass of fuel f0 has a drive which is
|
||
capable of constant fuel consumption K and an exhaust velocity
|
||
(specific impulse) E.
|
||
|
||
Final change in velocity after exhausting fuel, taking into account
|
||
relativity:
|
||
|
||
v|T = (c E D)/(c^2 + E^2 D^2)^(1/2),
|
||
|
||
where D = ln (m0 + f0) - ln m0.
|
||
|
||
|
||
MOMENTS OF INERTIA
|
||
|
||
Do you want these?
|
||
|
||
|
||
RINGWORLD PARAMETERS
|
||
(figures from _The Ringworld Role-Playing Game_)
|
||
|
||
Mass of Ringworld:
|
||
|
||
m = 2.1 x 10^27 kg.
|
||
|
||
Radius of Ringworld:
|
||
|
||
r = 1.5288 x 10^11 m.
|
||
|
||
Width of Ringworld:
|
||
|
||
w = 1.604 x 10^9 m.
|
||
|
||
Thickness of Ringworld:
|
||
|
||
t = 30 m.
|
||
|
||
Spin-induced "gravity" at surface of Ringworld:
|
||
|
||
a = 9.73 m/s^2.
|
||
|
||
Apparently angular size of portion of Ringworld floor as seen from
|
||
surface of Ringworld a central angle theta away:
|
||
|
||
phi = arctan [(s/r)/(4 sin theta)].
|
||
|
||
Tension stress induced in Ringworld by rotation:
|
||
|
||
sigma = m a/(2 pi w t).
|
||
|
||
|
||
ANALYSES OF ORBITS
|
||
|
||
A body falls from a distance r0 to a distance r from a central body
|
||
with mass M.
|
||
|
||
Accumulated deltavee:
|
||
|
||
deltav = [2 G M/(1/r - 1/r0)].
|
||
|
||
|
||
A planet with mass m is in orbit around a Sun with mass M. The
|
||
planet's orbit has eccentricity e and angular momentum l.
|
||
|
||
Energy associated with orbit:
|
||
|
||
E = [G^2 m^3 M^2/(2 l^2)] (e^2 - 1).
|
||
|
||
For an ellipse with semimajor axis a:
|
||
|
||
E = -G m M/(2 a).
|
||
|
||
|
||
VELOCITIES OF BODIES IN ORBIT
|
||
|
||
A planet is in orbit around a Sun with mass M. The following
|
||
represents the velocity v at a point when the planet is at a distance
|
||
r from the primary.
|
||
|
||
For a circle:
|
||
|
||
v = (G M/r)^2.
|
||
|
||
For an ellipse (with semimajor axis a):
|
||
|
||
v = [2 G M [1/r - 1/(2 a)]].
|
||
|
||
For a parabola:
|
||
|
||
v = (2 G M/r)^2.
|
||
|
||
For an hyperbola with eccentricity e and minimum planet-Sun distance
|
||
alpha:
|
||
|
||
v = [2 G M [1/r + (e - 1)/(2 alpha)]].
|
||
|
||
|
||
HAWKING RADIATION
|
||
|
||
For a black hole with mass m, total power output P is
|
||
|
||
P = k/m^2
|
||
|
||
where k is a constant, on the order of 10^34 kg m^2.
|
||
|
||
Blackbody thermodynamic temperature of black hole with mass m (sigma
|
||
is Stefan-Boltzmann constant):
|
||
|
||
T = [(c^4 k)/(16 pi sigma G^2)]^(1/4) (1/m).
|
||
|
||
For a black hole with initial mass m0, the amount of time required for
|
||
black hole to completely dissolve is
|
||
|
||
tau = c^2 m0^3/(3 k).
|
||
|
||
|
||
GRAVITATIONAL DISRUPTION
|
||
|
||
Total energy required to completely gravitational disrupt a uniform,
|
||
spherical body of mass M and radius R:
|
||
|
||
E = (9/15) G M^2/R.
|
||
|
||
|
||
That's most of the stuff I have right now. Please let me know if you
|
||
have any questions or problems with the above formulae. If you have
|
||
any other specific problems you'd like to have solutions to (I love to
|
||
solve problems), let me know and I'll see if I can work them out.
|
||
|