[ 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.