http://www.projectrho.com/rocket/AtomicRocketWiki.html
AtomicRockets
atomic rocketship of the space patrol
Acceleration due to gravity at Earth's suface //(m/////s)//\n''g~~0~~'' = 9.81 meters per second
Duration of the current burn //(s)//
Mass of everything that is not propulsion system, structural, or propellant //(kg)//\nCrew, oxygen, life-support, supplies, cargo, everything.
Mass of rocket's structure //(kg)//\nRocket's framework and hull.
Mass of the electrical power plant //(kg)//\nNaturally this is only large if the rocket utilized electrical propulsion.
Mass of the system that actually produced thrust //(kg)//
Equations
Distance from the the geometric figure's center to the circumference //(m)//
Distance from a geometric solid's base to its apex //(m)//
Distance to target //(m)//\n\nRemember that one Astronomical Unit is 1.49 x 10^^11^^ meters
Surface area of a circle //(m^^2^^)//\n''S~~circle~~'' = [[π]] × [[Radius]]^^2^^
Surface area of a sphere //(m^^2^^)//\n''S~~sphere~~'' = 4 × [[π]] × [[Radius]]^^2^^\n
Surface area of a cube //(m^^2^^)//\n''S~~cube~~'' = 6 × edge^^2^^
The volume of a sphere //(m^^3^^)//\n''V~~sphere~~'' = 4/3 × [[π]] × [[Radius]]^^3^^\n''V~~sphere~~'' = (4 × [[π]] × [[Radius]]^^3^^) / 3\n''V~~sphere~~'' ≈ 1.3333 × [[π]] × [[Radius]]^^3^^ //(approximately)//\n''V~~sphere~~'' ≈ 4.18879 × [[Radius]]^^3^^ //(approximately)//
Constant acceleration required for Brachistochrone trajectory of desired distance and duration //(m/////s^^2^^)//\n''A~~brachistochron~~'' = (4 × [[Distance]]) / BrachistochroneDuration^^2^^\n\nRemeber that\n1 g of acceleration = 9.81 m/s^^2^^\none-tenth g of acceleration = 0.981 m/s^^2^^\none one-hundredth g of acceleration = 0.0981 m/s^^2^^
Acceleration of rocket //(m/////s^^2^^)//\n''A'' = [[Thrust]] / CurrentMass\n''A'' = (MassFlow × ExhaustVelocity) / CurrentMass\n''A'' = (MassFlow × EarthGravity × SpecificImpulse) / CurrentMass\n\nRemeber that\n1 g of acceleration = 9.81 m/s^^2^^\none-tenth g of acceleration = 0.981 m/s^^2^^\none one-hundredth g of acceleration = 0.0981 m/s^^2^^
Change in velocity required for brachistochone trajectory //(m/////s)//\n\n''Δ~~vbrachistochrone~~'' = 2 × ( √( 2 × BrachistochroneAcceleration × ([[Distance]]/2)) )
Time required for a constant-acceleration rocket to accelerate up to the midpoint, do a skew-flip, and deaccelerate to the destination //(s)//\n\n''T'' = 2 × √( [[Distance]] / BrachistochroneAcceleration )\n\nPlease note this is for //constant//-acceleration rocket, rockets must gradually throttle back to do this since the [[Acceleration]] goes up as mass is lost due to propellant being expended.\n\nDivide time in seconds by 3600 for hours, 86,400 for days, 2,592,000 for //(30 day)// months, or 31,536,000 for years
Rocket's current maxium change in velocity capability //(m/////s)//\n''Δ~~vc~~'' = ExhaustVelocity × //ln//(CurrentMass / DryMass)\n''Δ~~vc~~'' = EarthGravity × SpecificImpulse × //ln//(CurrentMass / DryMass)
Instantaneous current mass of rocket //(kg)//\n''M~~i~~'' = [[Thrust]] / [[Acceleration]]\n''M~~i~~'' = (MassFlow × ExhaustVelocity) / [[Acceleration]]\n''M~~i~~'' = (MassFlow × EarthGravity × SpecificImpulse) / [[Acceleration]]
Rocket exhaust velocity //(m/////s)//\n''V~~e~~'' = EarthGravity × SpecificImpulse\n''V~~e~~'' = [[Thrust]] / MassFlow
Rocket's mass flow //(kg/////s)//\nmDot = PropellantMassBurnt / BurnDuration\nmDot = [[Thrust]] / (EarthGravity × SpecificImpulse)\nmDot = [[Thrust]] / ExhaustVelocity
Mass of propulsion system //(kg)//\n''M~~ps~~'' = PowerPlantMass + ThrusterMass
Mass of propellant expended in current burn //(kg)//\n''M~~pb~~'' = MassFlow × BurnDuration\n''M~~pb~~'' = ([[Thrust]] × BurnDuration) / (EarthGravity × SpecificImpulse)\n''M~~pb~~'' = ([[Thrust]] × BurnDuration) / ExhaustVelocity\n
Rocket thrust //(Newtons)// or //(kg m/////s^^2^^)//\n''F'' = CurrentMass × [[Acceleration]]\n''F'' = MassFlow × ExhaustVelocity\n''F'' = MassFlow × EarthGravity × SpecificImpulse\n''F'' = (PropellantMassBurnt × ExhaustVelocity) / BurnDuration
Specific Impulse //(s)//\n''Isp'' = ExhaustVelocity / EarthGravity\n''Isp'' = [[Thrust]] / (EarthGravity × MassFlow)
Energy in the thrust //(w)//\n''F~~p~~'' = (MassFlow × (ExhaustVelocity^^2^^)) / 2\n''F~~p~~'' = ([[Thrust]] × ExhaustVelocity ) / 2\n''F~~p~~'' = (PropellantMassBurnt × (ExhaustVelocity^^2^^)) / (2 × BurnDuration)
Number of stars within a spherical empire.\n''N~~stars~~'' = EmpireRadius^^3^^ × StarDensity\n\n//Note:// this equation will produce incorrect results if EmpireRadius is larger than 1000 ly. This is due to the fact that the galactic disc is only about 1000 ly thick in our region of the galaxy.
Number of stars hosting at least one human-habitable planet inside a spherical empire.\n''N~~hStars~~'' = EmpireRadius^^3^^ × HabitableStarDensity\n\n//Note:// this equation will produce incorrect results if EmpireRadius is larger than 1000 ly. This is due to the fact that the galactic disc is only about 1000 ly thick in our region of the galaxy.\n
V~~c~~: Velocity of the wave of new planets being colonized //(light-years/////y)//\n\nV~~c~~ = Δ~~r~~/ [(Δ/v~~s~~) + (1/α) //ln(//2α/γ//)//]\nV~~c~~ =ColonyRadialDistance / [(ColonyDistance/ColonyShipSpeed) + (1/PopulationGrowth) × //ln(//2 × PopulationGrowth/EmigrationRate//)//]\n\nOne can generally assume that ColonyRadialDistance = 0.7 × ColonyDistance and neglect travel time:\nV~~c~~ = 0.7αΔ / //ln(//2α/γ//)//\nV~~c~~ = 0.7 × ColonyDistance / //ln(//2 × PopulationGrowth/EmigrationRate//)//\n\n//Example:// If ColonyDistance=7.2 light years //(2.2 parsecs)//, PopulationGrowth=10^^-3^^, and EmigrationRate=10^^-4^^, then the colonization wave will travel at about 2×10^^-3^^ light-years per year //(5×10^^-4^^ parsecs per year)//\n\n<<<\nEquation is from Eric M. Jones, //Discrete calculations of interstellar migration and settlement// ( [[Icarus Volume 46, Issue 3|http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WGF-47315YW-1J6&_coverDate=06%2F30%2F1981&_alid=230714783&_rdoc=1&_fmt=&_orig=search&_qd=1&_cdi=6821&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=e5bfc8acc66d532619c0f9792743b838]] , June 1981, Pages 328-336.\n<<<
Δ: mean separation between colony sites //(light-years)//
Δ~~r~~ average radial distance traveled by colony ships //(i.e., distance as meaured from the center of the empire)// //(light-years)//
v~~s~~ Velocity of colony ships //(light-year/////y)//
NeutralZoneVolume * HabitableStarDensity
γ: Emigration rate //(percentage of current population ///// y)//\n\nA modest emigrationrate is 10^^-4^^/year //(0.0001)//. A draconian emigration rate is 10^^-2^^/year //(0.01)//, which was last seen in the great Irish emigration of the 1840's.\n
α: Population growth per year //(percentage of current population ///// y)//\n\nA modest population growth is 10^^-3^^/y or 0.1% annually. The United States is currently experiencing population growth of approximately 6×10^^-3^^/y or 0.6% annually. Some nations are crowding 3×10^^-2^^/y or 3.0% annually.
The "Neutral zone" between two spherical empires is defined as the lens-shaped intersection of the two spheres. A neutral zone is typically populated by spies, smugglers, covert battlefleets intent on causing boarder incidents, and planets named "Casablanca".
Volume of the Neutral zone //(light-year^^3^^)//\n\n''V~~nz~~'' = [[π]](R+r-d)^^2^^ (d^^2^^ + 2dr - 3r^^2^^ + 2dR + 6rR - 3R^^2^^) / 12d\n''V~~nz~~'' = ([[π]] × ( EmpireRadius1 + EmpireRadius2 - EmpireSeparation)^^2^^ × (EmpireSeparation^^2^^ + (2 × EmpireSeparation × EmpireRadius2) - (3 × EmpireRadius2 ^^2^^) + (2 × EmpireSeparation × EmpireRadius1 ) + (6 × EmpireRadius2 × EmpireRadius1) - (3 × EmpireRadius1^^2^^))) / (12 × EmpireSeparation)
Radius of a spherical interstellar empire//(light-years)//\n''R~~ly~~'' = ^^3^^√(EmpireStars × 97)\n''R~~ly~~'' = ^^3^^√(EmpireHabitableStars × 464)\n''R~~ly~~'' = ColonyVelocity × ColonizationDuration\n\nNote that if the equation yields a value for R~~ly~~ larger than 1000 ly it will be inaccurate. This is due to the fact that the galactic disc is only about 1000 ly thick in our region of the galaxy.\n\n<<<\nEquations are based upon an analysis of the [[HabCat database|http://www.projectrho.com/smap06.html#new135]] compiled by [[Jill Tarter and Margaret Turnbull|http://www.astrobio.net/news/article436.html ]].\n<<<
Duration of the colonization era //(y)//
Volume of the galaxy //(light-year^^3^^)//\n''V~~galaxy~~'' ≈ 5.65x10^^12^^ light-year^^3^^\n\nThis is difficult to get a value for that everybody agrees on. Do your own research.\n
Number of empires currently existing in the galaxy.\nThis can be approimated by using different guesses for various values in the [[Drake|http://www.activemind.com/Mysterious/Topics/SETI/drake_equation.html]] [[Equation|http://zebu.uoregon.edu/~imamura/122/mar15/mar15.html]].
The [["number density"|http://en.wikipedia.org/wiki/Number_density]] of stars in our galaxy //(stars/////ly^^3^^)//\n\nThere are approximately 0.01 stars per cubic light year.\n<<<\nValue is based upon an analysis of the [[HabCat database|http://www.projectrho.com/smap06.html#new135]] compiled by [[Jill Tarter and Margaret Turnbull|http://www.astrobio.net/news/article436.html ]].\n<<<
The [["number density"|http://en.wikipedia.org/wiki/Number_density]] of stars in our galaxy hosting a human-habitable planet //(stars/////ly^^3^^)//\n\nThere are approximately 0.0022 human-habitable stars per cubic light year.\n<<<\nValue is based upon an analysis of the [[HabCat database|http://www.projectrho.com/smap06.html#new135]] compiled by [[Jill Tarter and Margaret Turnbull|http://www.astrobio.net/news/article436.html ]].\n<<<
The average distance between stars in the galaxy //(light-years)//\n''Δ~~star~~'' = ^^3^^√(1/StarDensity)\n
The average distance between habitable stars in the galaxy //(light-years)//\n''Δ~~star~~'' = ^^3^^√(1/HabitableStarDensity)\n
Distance between the two empires boarding the neutral zone //(light-years)//\nSee AverageEmpireSeparation
Age of the galaxy //(y)//\nApproximately 10^^10^^ years
Aveage distance between empires //(light-years)//\n\n''D~~empire~~'' = ^^3^^√(GalacticVolume/ EmpireNumber )\n\nCarl Sagan came up with:\n''D~~empire~~'' = StarSeparation * ^^3^^√(GalacticAge / (ImperialPercent × EmpireLifespan))
''N~~nzStars~~'' = NeutralZoneVolume × StarDensity
Fraction of all stars in the galaxy which are the homeworlds of races which are at least advanced enough to found empires. If you assume that Earth is approximately advanced enough, and Earth is average, then \n\nI = (HabitableStarDensity / StarDensity) / 2\nI = 0.11\n\nCar Sagan's estimate is I = 0.01\n\n//divided by 2 since if Earth is average, half will be not advanced enough and half will be advanced enough//
Average lifespan of an empire //(y)//\n\nThis is anybody's guess.\nCarl Sagan's estimate is 10^^5^^ years. This might be overly generous if empires are prone to entering a [[Singularity|http://en.wikipedia.org/wiki/Technological_singularity]].
Distance between two stars.\n''Δ*'' = √((X~~1~~ - X~~2~~)^^2^^ + (Y~~1~~ - Y~~2~~)^^2^^ + (Z~~1~~ - Z~~2~~)^^2^^)\n\nwhere the start star is at cartesian coordinates X~~1~~,Y~~1~~,Z~~1~~ and the destination star is at X~~2~~,Y~~2~~,Z~~2~~
Rho is distance\n''ρ'' = distance between Sol and the star
Theta is decimal Declination\n''Θ'' = ( //abs(//DeclinationDegrees//)// + DeclinationMinutes/60 + DeclinationSeconds/3600 ) * //sign(//DeclinationDegrees//)//
*SurfaceAreaCircle\n*SurfaceAreaCone\n*SurfaceAreaCube\n*SurfaceAreaSphere\n*VolumeSphere
*GeneralEquations\n*EmpireEquations\n*GeometricEquations\n*RocketryEquations
Z = [[ρ]] × //sin(//[[Θ]]//)//
Phi is decimal Right Ascension\n''Φ'' = ( RightAscensionHours × 15 ) + ( RightAscensionMinutes × 0.25 ) + ( RightAscensionSeconds × 0.0041666 )
R~~v~~ = [[ρ]] × //cos(// [[Θ]] //)//
//epoch 1950.0://\nXg = -(0.0672 × [[X]]) - (0.8727 × [[Y]]) - (0.4835 × [[Z]])\n\n//epoch 2000.0://\nXg = -(0.0550 × [[X]]) -(0.8734 × [[Y]]) - (0.4839 × [[Z]])
Base of the natural logarithms.\ne ≈ 2.71828 //(approximately)//
Pi is the ratio of the circumference of a circle to its diameter.\n''π'' ≈ 3.14159 //(approximately)//