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 Useful formulae and constants (IE may not work) (Read 11132 times)
 abyssoft YaBB Administrator I love YaBB 1G - SP1! Posts: 302 Useful formulae and constants (IE may not work) 01/15/12 at 11:53:05   Angular Frequency     ω=2π/T=2πf=|v|/|r|     ω is the angular frequency or angular speed (radians per second) T is the period (seconds) f is the ordinary frequency (hertz,sometimes symbolised with ν) v is the tangential velocity of a point about the axis of rotation (meters per second) r is the radius of rotation (meters) Back to top « Last Edit: 03/12/12 at 16:44:19 by abyssoft »     IP Logged
 abyssoft YaBB Administrator I love YaBB 1G - SP1! Posts: 302 Re: Useful formulae and constants Reply #1 - 01/15/12 at 12:10:59   Oblateness Constant (simple)                 q ≡  (aω^2)/g=aω^2  a^2/GM=(a^3 ω^2)/GM   a is equatorial radius (meters) ω is the angular frequency or angular speed (radians per second) g is gravitational acceleration(meters per second) G is the graviational constant 6.67384×10^(-11)  m^3 kg^(-1) s^(-2) M is mass of body (kilograms) This does not take into account differentiation.  To fudge this on can take M and multiply against either a random number between 0.8 and 1.2 or mulitply against a system wide value to represent general differentiation. Back to top « Last Edit: 03/12/12 at 16:43:34 by abyssoft »     IP Logged
 abyssoft YaBB Administrator I love YaBB 1G - SP1! Posts: 302 Re: Useful formulae and constants Reply #2 - 01/15/12 at 16:20:45   Roche limit advanced                     d≈2.423R(ρM/ρm)^(1/3)·{[(1+m/3M)+1/3q(1+m/M)]/(1-q)}^(1/3)[/size]   d is distance from center of primary ρ_M  is density of primary ρ_m  is density of satillite M is mass in of primary m is mass in of satillite q is the oblateness constant (also listed as c/R) [/size] Back to top « Last Edit: 03/12/12 at 16:42:23 by abyssoft »     IP Logged
 abyssoft YaBB Administrator I love YaBB 1G - SP1! Posts: 302 Re: Useful formulae and constants Reply #3 - 01/15/12 at 19:17:54   Orbital Period                     P=2π√(a^3/G(M+m))   P is period of orbit (seconds) a is separation distance (meters) M is mass in of primary m is mass in of satillite G is the graviational constant 6.67384×10^(-11)  m^3 kg^(-1) s^(-2) Back to top « Last Edit: 03/12/12 at 16:41:39 by abyssoft »     IP Logged
 abyssoft YaBB Administrator I love YaBB 1G - SP1! Posts: 302 Re: Useful formulae and constants Reply #4 - 01/15/12 at 19:20:07   Minimum Rotational Period         P=2π√(r^3/GM)   P is period of orbit (seconds) r is equatorial radius (meters) M is mass of body (kilograms) G is the gravitational constant 6.67384x10-11 m^3 kg^(-1) s^(-2) Back to top « Last Edit: 03/12/12 at 16:40:45 by abyssoft »     IP Logged
 abyssoft YaBB Administrator I love YaBB 1G - SP1! Posts: 302 Re: Useful formulae and constants Reply #5 - 01/15/12 at 22:54:33   Synchronous Orbit                     r=(GM/ω^2)^(1/3)   M is mass of body (kilograms) G is the gravitational constant 6.67384x10-11 m^3 kg^(-1) s^(-2)   r is the elevation from the center of the body (meters) ω is the angular frequency or angular speed radians per second Back to top « Last Edit: 03/12/12 at 16:40:10 by abyssoft »     IP Logged
 abyssoft YaBB Administrator I love YaBB 1G - SP1! Posts: 302 Re: Useful formulae and constants Reply #6 - 01/15/12 at 22:55:08   Synchronous Orbit Velocity         v=ωr   v is the velocity of the synchronous orbit (meters per second) r is the elevation from the center of the body (meters) ω is the angular frequency or angular speed (radians per second) Back to top « Last Edit: 03/12/12 at 16:39:18 by abyssoft »     IP Logged
 abyssoft YaBB Administrator I love YaBB 1G - SP1! Posts: 302 Re: Useful formulae and constants (IE may not work Reply #7 - 01/16/12 at 20:13:35   Effective Planet Temperature         Teff=((L☉ (1-A))/(16πσa^2 ))^(1/4)   Teff  is the effective surface temperature not accounting for greenhouse effect (kelvin) L☉ is the luminosity in solar units and must be converted to watts L☉∙3.846×10^26  W A is the geometric albedo (earth = 0.367) σ is the Stefan-Boltzmann constant 5.670400×10^(-8) J s^(-1) m^(-2) K^(-4) a is the semi-major axis(meters)   To get minimum Teff and maximum Teff adjust to perihelion and apohelion   This does not take into account any greenhouse effect factor,H,to add in H use the following modification to the T_eff  formula   Twgh=Teff(1+0.15∙H/|H|∙|H|^(1/2))   Twgh is the temperature factoring in with the greenhouse effect H is the Green House Factor (earth = 1).   It is not unusual for H to be a negative value for bodies who either have   a highly reflective surface or a thin conductive atmosphere, Enceladus and Mars   are prime examples.     For gas planets that are in excess of 12 Mjup  black body value for planet should be included at equatorial radius as planet will most   likely still be radiating heat from formation. Back to top « Last Edit: 03/12/12 at 16:38:23 by abyssoft »     IP Logged
 abyssoft YaBB Administrator I love YaBB 1G - SP1! Posts: 302 Re: Useful formulae and constants (IE may not work Reply #8 - 01/16/12 at 23:10:39   Luminosity Mass Relationship   L=kM^n   L is the luminosity of the stellar object k is multiplier factor M is the mass of the stellar object in Solar masses n is the power factor   k and n vary according to M M≤0.43;k=0.23,n=2.3 M≤2.00 and M>0.43;k=1,n=4 M≤20.0 and M>2.00;k=1.5,n=3.5 M>20.0;k=1;n=4-M/(44+(M/10.25)^2 ) Back to top « Last Edit: 03/12/12 at 17:12:42 by abyssoft »     IP Logged
 abyssoft YaBB Administrator I love YaBB 1G - SP1! Posts: 302 Re: Useful formulae and constants (IE may not work Reply #9 - 01/16/12 at 23:11:19   Volume of an oblate spheroid         V=4/3 πr_e^2 r_p   V is the volume of the  body r_e  is the equatorial radius r_p  is the polar radius r_e (1-q) Back to top « Last Edit: 03/12/12 at 17:14:53 by abyssoft »     IP Logged
 abyssoft YaBB Administrator I love YaBB 1G - SP1! Posts: 302 Re: Useful formulae and constants (IE may not work Reply #10 - 01/16/12 at 23:15:29   Hill sphere                           r_hill≈a(m/3M)^(1/3)   a is the semi-major axis of the orbit M is the mass of the primary m is the mass of the secondary   Maximum distance for stable retrograde orbit     d=(((δ_s+φ)/2))/3 r_hill   Maximum distance for stable prograde orbit   d=φ/3 r_hill   d is distance to limit δ_s  is the silver ratio 1+2^(1/2) φ is the golden ratio (1 + 5^(1/2))/2 r_hill  is the hill sphere radius Back to top « Last Edit: 03/12/12 at 17:22:39 by abyssoft »     IP Logged
 abyssoft YaBB Administrator I love YaBB 1G - SP1! Posts: 302 Re: Useful formulae and constants (IE may not work Reply #11 - 01/17/12 at 13:58:35   Gravitational dominance range of a perturbing body        Inner Reach                                                                  d_inn=a((1-e)-n_inner(m/3M)^(1/3) )        Outer Reach                                                                  d_out=a((1+e)+n_outer(m/3M)^(1/3) )   d is the distance inward(inn) or outward(out) a is the semi-major axis of the bodies,for the star use radius of star for outer reach e is the eccentricity of the orbiting body,for star use eccentricity of 0 n is the scalar modifier more on this value can be found at Jones et. al. (2006)     Values for n and means to calculate were refined for accuracy and better curve fitting; I was assisted by Tony on this.     M is the mass of the primary m is the mass of the secondary Back to top « Last Edit: 03/12/12 at 22:35:27 by abyssoft »     IP Logged
 abyssoft YaBB Administrator I love YaBB 1G - SP1! Posts: 302 Re: Useful formulae and constants (IE may not work Reply #12 - 01/19/12 at 12:25:19   Interplanetary region can have belt and/or dwarf planet when BD_bool is true                    BD_bool = ( d_inn [outerbody] - d_out [innerbody] - d_inn [innerbody] ) > 0   *commentary* Minimum mass of dwarf planet can be based on the smallest known body Saturn I (Mimas) to be in hydrostatic equilibrium with a mass of 3.75×10^19  kg. However, it may be slightly lower if composition was structurally less rigid.   */commentary*   When BD_bool is true, Maximum expanse of a belt or range for possible stable orbits of a dwarf planet   Expanse_middle = d_inn [outerbody]-d_out [innerbody] Expanse_width = 1.5 (d_inn [outerbody]-d_out [innerbody]-d_inn [innerbody] ) Back to top « Last Edit: 03/12/12 at 22:51:32 by abyssoft »     IP Logged
 abyssoft YaBB Administrator I love YaBB 1G - SP1! Posts: 302 Re: Useful formulae and constants (IE may not work Reply #13 - 01/19/12 at 14:53:37   Concerning Maximum mass of dwarf planet or intraexpanse planet   Stable orbits for bodies within an expanse are     1) typically not in resonance with the outer planet;   1a) in the case of those beyond the the last major planet, resonances with the planet typical ensure stability up to 1:4 1b) weaker resonances may be present but may mearly be coincidental or transient.   2) e is frequently a value between the e of the inner and outer planet, as outside the range can lead to overlaping gravitational dominance zones, which would lead to eventual destablization of the orbit.     3) will not be stable when at the edges (+/-10%) of the expanse unless in resonance with inner planet and inner planet is significantly more massive then outer. (Empirical testing through simulations)   To find the maximum mass that would be stable in the system within the selected parameters use one the following.   The values for a, e, and either d_inn or d_out must be selected/determined before solving for m.     m = -(3( a * e - a + d_inn )^3)/(a^3 n^3 M)   m = -(3( a * e + a - d_out )^3)/(a^3 n^3 M)   refer to the variables in Gravitational dominance range of a perturbing body for explinations Back to top « Last Edit: 03/12/12 at 22:56:02 by abyssoft »     IP Logged
 abyssoft YaBB Administrator I love YaBB 1G - SP1! Posts: 302 Re: Useful formulae and constants (IE may not work Reply #14 - 01/19/12 at 14:54:49   Kirkwood gaps,   Kirkwood peaks   Continued analysis,experimentation, along with the following paper I found earlier today (http://www.fisica.edu.uy/~gallardo/marte12/mars1to2.html ) has yielded that resonances with the inner planet strengthen the population of SSSB; leading to Kirkwood peaks.   There also seems to be some impact by the resonances with both the inner reach and the outer reaches.   I'm investigating the resonances with the Hill sphere ranges.   Still working on these will update as I get these nailed down a bit better, or discover more.     The deeper analysis is taking quite a bit of time as I run both sims and statistical sequences. Back to top « Last Edit: 01/31/12 at 20:24:02 by abyssoft »     IP Logged