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General >> Discussion >> Useful formulae and constants (IE may not work)
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Message started by abyssoft on 01/15/12 at 11:53:05

Title: Useful formulae and constants (IE may not work)
Post by abyssoft on 01/15/12 at 11:53:05

Angular Frequency

http://www.texify.com/img/%5CLARGE%5C%21%5Comega%3D%5Cfrac%7B2%5Cpi%7D%7BT%7D%3D2%5Cpi%20f%3D%5Cfrac%7B%5Cleft%7C%20v%5Cright%7C%7D%7B%5Cleft%7C%20r%5Cright%7C%7D.gif
[ch969]=2[ch960]/T=2[ch960]f=|v|/|r|

[ch969] is the angular frequency or angular speed (radians per second)
T is the period (seconds)
f is the ordinary frequency (hertz,sometimes symbolised with [ch957])
v is the tangential velocity of a point about the axis of rotation (meters per second)
r is the radius of rotation (meters)

Title: Re: Useful formulae and constants
Post by abyssoft on 01/15/12 at 12:10:59

Oblateness Constant (simple)            

http://www.texify.com/img/%5CLARGE%5C%21q%20%5Cequiv%20%5Cfrac%7Ba%5Comega%5E2%7D%7Bg%7D%20%3D%20a%5Comega%5E2%5Cfrac%7Ba%5E2%7D%7BGM%7D%20%3D%20%5Cfrac%7Ba%5E3%5Comega%5E2%7D%7BGM%7D.gif
q [ch8801]  (a[ch969]^2)/g=a[ch969]^2  a^2/GM=(a^3 [ch969]^2)/GM

a is equatorial radius (meters)
[ch969] 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.

Title: Re: Useful formulae and constants
Post by abyssoft on 01/15/12 at 16:20:45

Roche limit advanced                  

http://www.texify.com/img/%5CLARGE%5C%21d%20%5Capprox%202.423R%20%20%5Cleft%28%5Cfrac%7B%5Crho_M%7D%7B%5Crho_m%7D%5Cright%29%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%20%5Cleft%28%20%5Cfrac%7B%20%5Cleft%281%2B%5Cfrac%7Bm%7D%7B3M%7D%20%5Cright%29%2B%5Cfrac%7B1%7D%7B3%7Dq%5Cleft%281%2B%5Cfrac%7Bm%7D%7BM%7D%5Cright%29%20%7D%7B1-q%7D%20%5Cright%29%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D.gif
d[ch8776]2.423R([ch961]M/[ch961]m)^(1/3)·{[(1+m/3M)+1/3q(1+m/M)]/(1-q)}^(1/3)[/size]

d is distance from center of primary
[ch961]_M  is density of primary
[ch961]_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]

Title: Re: Useful formulae and constants
Post by abyssoft on 01/15/12 at 19:17:54

Orbital Period                  

http://www.texify.com/img/%5CLARGE%5C%21P%3D2%5Cpi%5E2%5Csqrt%7B%5Cfrac%7Ba%5E3%7D%7BG%5Cleft%28M%2Bm%5Cright%29%7D%7D.gif
P=2[ch960][ch8730](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)

Title: Re: Useful formulae and constants
Post by abyssoft on 01/15/12 at 19:20:07

Minimum Rotational Period      

http://www.texify.com/img/%5CLARGE%5C%21P%3D2%5Cpi%5E2%5Csqrt%7B%5Cfrac%7Br%5E3%7D%7BGM%7D%7D.gif
P=2[ch960][ch8730](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)

Title: Re: Useful formulae and constants
Post by abyssoft on 01/15/12 at 22:54:33

Synchronous Orbit                  

http://www.texify.com/img/%5CLARGE%5C%21r%3D%5Csqrt%5B3%5D%7B%5Cfrac%7BGM%7D%7B%5Comega%5E2%7D%7D.gif
r=(GM/[ch969]^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)
[ch969] is the angular frequency or angular speed radians per second


Title: Re: Useful formulae and constants
Post by abyssoft on 01/15/12 at 22:55:08

Synchronous Orbit Velocity      

http://www.texify.com/img/%5CLARGE%5C%21v%3D%5Comega%20r.gif
v=[ch969]r

v is the velocity of the synchronous orbit (meters per second)
r is the elevation from the center of the body (meters)
[ch969] is the angular frequency or angular speed (radians per second)

Title: Re: Useful formulae and constants (IE may not work
Post by abyssoft on 01/16/12 at 20:13:35

Effective Planet Temperature      

http://www.texify.com/img/%5CLARGE%5C%21T_%7Beff%7D%3D%7B%5Cleft%28%5Cfrac%7BL_%5Codot%5Cleft%281-A%5Cright%29%7D%7B16%5Cpi%5Csigma%20a%5E2%7D%5Cright%29%7D%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D.gif
Teff=((L[ch9737] (1-A))/(16[ch960][ch963]a^2 ))^(1/4)

Teff  is the effective surface temperature not accounting for greenhouse effect (kelvin)
L[ch9737] is the luminosity in solar units and must be converted to watts L[ch9737][ch8729]3.846×10^26  W
A is the geometric albedo (earth = 0.367)
[ch963] 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

http://www.texify.com/img/%5CLARGE%5C%21T_%7Bwgh%7D%3DT_%7Beff%7D%5Cleft%281%2B0.15%5Ccdot%5Cfrac%7B%5Cmathbb%7BH%7D%7D%7B%5Cleft%7C%5Cmathbb%7BH%7D%5Cright%7C%7D%5Ccdot%7B%5Cleft%7C%5Cmathbb%7BH%7D%5Cright%7C%7D%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%5Cright%29.gif
Twgh=Teff(1+0.15[ch8729]H/|H|[ch8729]|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.

Title: Re: Useful formulae and constants (IE may not work
Post by abyssoft on 01/16/12 at 23:10:39

Luminosity Mass Relationship

http://www.texify.com/img/%5CLARGE%5C%21L%3DkM%5En%5C%5B%5Cbegin%7Bmatrix%7D%5Ctext%7Bk%20and%20n%20vary%20according%20to%7D%20%26%20M%20%5C%5C%20%7Bk%20%3D%200.23%2C%20n%20%3D%202.3%7D%20%26%20%7B%5Cleft%28%200.00%2C%200.43%20%5Cright%5D%7D%20%5C%5C%20%7Bk%20%3D%201.00%2C%20n%20%3D%204%7D%20%26%20%7B%5Cleft%280.43%2C%202.00%5Cright%5D%7D%20%5C%5C%20%7Bk%20%3D%201.00%2C%20n%20%3D%203.50%7D%20%26%20%7B%5Cleft%282.00%2C%2020.0%5Cright%5D%7D%20%5C%5C%20%7Bk%20%3D%201.00%2C%20n%20%3D%20%7B4-%5Cfrac%7BM%7D%7B44%2B%5Cleft%28%5Cfrac%7BM%7D%7B10.25%7D%5Cright%29%5E%7B2%7D%20%7D%7D%20%7D%20%26%20%7B%5Cleft%2820.0%2C%5Cquad%5Cinf%5Cquad%5Cright%5D%7D%20%5Cend%7Bmatrix%7D%5C%5D.gif
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[ch8804]0.43;k=0.23,n=2.3
M[ch8804]2.00 and M>0.43;k=1,n=4
M[ch8804]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 )

Title: Re: Useful formulae and constants (IE may not work
Post by abyssoft on 01/16/12 at 23:11:19

Volume of an oblate spheroid      

http://www.texify.com/img/%5CLARGE%5C%21V%20%3D%20%5Cfrac%7B4%7D%7B3%7D%5Cpi%20r_e%5E2%20r_p.gif
V=4/3 [ch960]r_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)

Title: Re: Useful formulae and constants (IE may not work
Post by abyssoft on 01/16/12 at 23:15:29

Hill sphere                        

http://www.texify.com/img/%5CLARGE%5C%21r_%7B%5Cnormalsize%7Bhill%7D%7D%5Capprox%20a%5E3%5Csqrt%7B%5Cfrac%7Bm%7D%7B3M%7D%7D.gif
r_hill[ch8776]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

http://www.texify.com/img/%5CLARGE%5C%21d%3D%5Cfrac%7B%5Cleft%28%5Cfrac%7B%5Cdelta_s%2B%5Cvarphi%7D%7B2%7D%5Cright%29%7D%7B3%7D%20r_%7B%5Cnormalsize%7Bhill%7D%7D.gif
d=((([ch948]_s+[ch966])/2))/3 r_hill

Maximum distance for stable prograde orbit

http://www.texify.com/img/%5CLARGE%5C%21d%3D%5Cfrac%7B%5Cvarphi%7D%7B3%7D%20r_%7B%5Cnormalsize%7Bhill%7D%7D.gif
d=[ch966]/3 r_hill

d is distance to limit
[ch948]_s  is the silver ratio 1+2^(1/2)
[ch966] is the golden ratio (1 + 5^(1/2))/2
r_hill  is the hill sphere radius

Title: Re: Useful formulae and constants (IE may not work
Post by abyssoft on 01/17/12 at 13:58:35

Gravitational dominance range of a perturbing body

     Inner Reach                  http://www.texify.com/img/%5CLARGE%5C%21d_%7B%5Cnormalsize%7Binn%7D%7D%3Da%5Cleft%28%20%5Cleft%281-e%5Cright%29%20-%20n_%7B%5Cnormalsize%7Binner%7D%7D%5Cleft%28%5Cfrac%7Bm%7D%7B3M%7D%5Cright%29%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%20%5Cright%29.gif
                                               d_inn=a((1-e)-n_inner(m/3M)^(1/3) )

     Outer Reach                  http://www.texify.com/img/%5CLARGE%5C%21d_%7B%5Cnormalsize%7Bout%7D%7D%3Da%5Cleft%28%20%5Cleft%281%2Be%5Cright%29%20%2B%20n_%7B%5Cnormalsize%7Bouter%7D%7D%5Cleft%28%5Cfrac%7Bm%7D%7B3M%7D%5Cright%29%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%20%5Cright%29.gif
                                               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.

http://www.texify.com/img/%5CLARGE%5C%21n_%7B%5Cnormalsize%7Binner%7D%7D%3D-14.38740251%20e%5E3%2B%2012.97628237%20e%5E2-%20%203.76153548%20e%20%2B%203.00000000.gif
http://www.texify.com/img/%5CLARGE%5C%21n_%7B%5Cnormalsize%7Bouter%7D%7D%3D53.06031011%20e%5E3%20-%2064.43502303%20e%5E2%20%2B%2030.80206188%20e%20%2B%203.00000000.gif

M is the mass of the primary
m is the mass of the secondary

Title: Re: Useful formulae and constants (IE may not work
Post by abyssoft on 01/19/12 at 12:25:19

Interplanetary region can have belt and/or dwarf planet when BD_bool is true

     http://www.texify.com/img/%5CLARGE%5C%21%7BBD%7D_%7Bbool%7D%3D%5Cleft%28d_%7Binn%7D_%7B%5Cnormalsize%7Bouterbody%7D%7D%20-%20d_%7Bout%7D_%7B%5Cnormalsize%7Binnerbody%7D%7D%20-%20d_%7Binn%7D_%7B%5Cnormalsize%7Binnerbody%7D%7D%5Cright%29%20%3E%200.gif
           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]
http://www.texify.com/img/%5CLARGE%5C%21%7BExpanse%7D_%7Bmiddle%7D%3D%5Cleft%28d_%7Binn%7D_%7B%5Cnormalsize%7Bouterbody%7D%7D%20-%20d_%7Bout%7D_%7B%5Cnormalsize%7Binnerbody%7D%7D%20%5Cright%29.gif
Expanse_width = 1.5 (d_inn [outerbody]-d_out [innerbody]-d_inn [innerbody] )
http://www.texify.com/img/%5CLARGE%5C%21%7BExpanse%7D_%7Bwidth%7D%3D1.5%5Cleft%28d_%7Binn%7D_%7B%5Cnormalsize%7Bouterbody%7D%7D%20-%20d_%7Bout%7D_%7B%5Cnormalsize%7Binnerbody%7D%7D%20-%20d_%7Binn%7D_%7B%5Cnormalsize%7Binnerbody%7D%7D%5Cright%29.gif

Title: Re: Useful formulae and constants (IE may not work
Post by abyssoft on 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.

http://www.texify.com/img/%5CLARGE%5C%21m%3D-%5Cfrac%7B3%7B%5Cleft%28ae-a%2Bd_%7Binn%7D%5Cright%29%7D%5E%7B3%7D%7D%7Ba%5E3n%5E3M%7D.gif
m = -(3( a * e - a + d_inn )^3)/(a^3 n^3 M)
http://www.texify.com/img/%5CLARGE%5C%21m%3D-%5Cfrac%7B3%7B%5Cleft%28ae%2Ba-d_%7Bout%7D%5Cright%29%7D%5E%7B3%7D%7D%7Ba%5E3n%5E3M%7D.gif
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

Title: Re: Useful formulae and constants (IE may not work
Post by abyssoft on 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.

Title: Re: Useful formulae and constants (IE may not work
Post by abyssoft on 01/20/12 at 14:40:31

Minimum time to tide lock with parent body
     T[ch8776](a^6 [ch969]IQ)/(3Gm_p^2 k_2 R^5 )
[ch969] is the initial rotation period of the primary body in rad/s
a is the semi-major axis of the secondary body
G is the gravational constant
m_p  is the mass of the primary body
m_s  is the mass of the secondary body
R is the radius of the secondary body
I is the moment of inertial for the secondary body
     I[ch8776]0.4m_s R^2
Q is the dissipation function of the secondary body
(typically between 50 and 500)  see http://en.wikipedia.org/wiki/Fluctuation_theorem#Dissipation_function for equation
     k_2[ch8776]1.5/(1+(19[ch956]/2[ch961]gR))
     [ch961] is the density of the secondary body
     [ch956] is the rigidity of the secondary body in terms of crushing pressure;
     This can be taken as 3×10^10  Nm^(-2) for rocky objects and 4×10^9  Nm^(-2)  
     for icy ones as an estimation,if you want to refine determine what pressure
     is needed per square meter to deform the normal crystal lattice
     of the substance averaged over the composition of the body.

g is the surface gravity of the secondary body
     g[ch8776](Gm_s)/(R_V^2 )
     R_v  is the mean volumetric radius of the secondary body

I believe this to be related to Obliquity Erosion when a planet looses it tilt over time but more research is needed.

Due to complexity I'm going to go about this one a bit differently for the image

http://www.texify.com/img/%5CLARGE%5C%21%5Cbegin%7Bgather%7Dg%5Capprox%5Cfrac%7BGm_%7B%5Cnormalsize%7Bs%7D%7D%7D%7BR_%7B%5Csmall%7BV%7D%7D%5E2%7D%20%5C%5C%20%5Cbullet%5Ccirc%5Cbullet%5Ccirc%5Cbullet%5Ccirc%5Cbullet%5Ccirc%5Cbullet%5Ccirc%5Cbullet%5Ccirc%20%5C%5C%20%7Bk_%7B%5Cnormalsize%7B2%7D%7D%5Capprox%5Cfrac%7B1.5%7D%7B%5Cleft%281%2B%5Cfrac%7B19%5Cmu%7D%7B2%5Crho%20gR%7D%5Cright%29%7D%7D%20%5C%5C%20%5Cbullet%5Ccirc%5Cbullet%5Ccirc%5Cbullet%5Ccirc%5Cbullet%5Ccirc%5Cbullet%5Ccirc%5Cbullet%5Ccirc%20%5C%5C%20%7BT%5Capprox%5Cfrac%7B%5Comega%20a%5E6IQ%7D%7B3G%7Bm_%7B%5Cnormalsize%7Bp%7D%7D%5E2%7Dk_2R%5E5%7D%20%5Cend%7Bgather%7D.gif

Combined as 1 Formula

http://www.texify.com/img/%5CLARGE%5C%21T%5Capprox%5Cfrac%7B%5Comega%20a%5E6IQ%7D%7B3G%7Bm_%7B%5Cnormalsize%7Bp%7D%7D%5E2%7DR%5E5%7B%5Cleft%28%5Cfrac%7B1.5%7D%7B%5Cleft%5B1%2B%5Cfrac%7B19%5Cmu%7D%7B2%20%5Crho%20R%20%5Cleft%7B%5Cfrac%7BGm_%7B%5Cnormalsize%7Bs%7D%7D%7D%7BR_%7B%5Csmall%7BV%7D%7D%5E2%7D%5Cright%7D%20%7D%5Cright%5D%7D%5Cright%29%7D%20%5C%5C%20.gif

Simplified
http://www.texify.com/img/%5CLARGE%5C%21T%5Capprox%5Cfrac%7B0.222222%20a%5E6%20I%20Q%20%5Comega%20%5Cleft%281%2B%5Cfrac%7B19%20R_v%5E2%5Cmu%7D%7B2%20G%20m_s%20%5Crho%20R%7D%5Cright%29%7D%7BG%20m_p%5E2%20R%5E5%7D.gif

Title: Re: Useful formulae and constants (IE may not work
Post by abyssoft on 03/13/12 at 13:22:01

Updated the posts with images still working on the Kirkwood gaps and peaks, the analysis is take a bit longer then I had hoped but I'm trying to make sure I have at least a 5 sigma confidence.

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