Barycentres in multiple (more than binary) systems: Spica

[Malenfant]

I'm getting kinda stuck trying to figure out the orbital configuration of stars in a multiple star system.

The system in question is Spica (the A and B components are accurate, the others are suspected to exist but there's little data on them so the data for those are mostly extrapolated/fictional). So far I've got this data (mass in solar mases, luminosity in Sols, Radius in Sols, Distance in AU):

Code: Select all

   Mass     Lum     Rad     Dist
A  11       13400   7.8        -
B   7        1700      4        0.12
C   6         900      3          4
D   4         250     2.5        40
E  0.6       0.08     0.54     10000


What I'm assuming is that there are nested barycentres here: I calculate the AB pair barycentre. Then I treat those two stars as a single mass to calculate where the AB-C barycentre is. I do the same to calculate where the ABC-D barycentre is. And again for the ABCD-E barycentre.

Is this how you're supposed to calculate where the barycentres are?

One thing I'm not sure about is if the separations shown are the distances between each star and A, or each star and its local barycenter. Apparently the formula to calculate a barycentre is:

r1 = rtot(m2/(m1+m2)) where r1 is the distance from the centre of the more massive star, rtot is the separation between the two stars, and m1 and m2 are the masses of the two stars. I'm using the distances shown as the rtot distance in this formula, is that what one is supposed to do?

If I do that, then I get these distances:

AB Barycentre = 0.047 AU from A
AB-C Barycentre = 1 AU from AB Bary
ABC-D Bary = 5.7143 AU from ABC Bary
ABCD-E Bary = 209 AU from ABCD Bary

The next problem of course is "how do I show this in Celestia"? Is it possible to 'nest' stellar barycentres like this yet? And how do you calculate the orbital periods of the stars around their barycentres?

[t00fri]

Malenfant,

the basic problem I see with your Spica system is that the gravitational multibody movement is getting really tough in the case that the involved masses are of similar magnitude. This is happens in your Spica setup (except for star E). So the orbits will not be simple ellipses anymore, since the gravitational pull from the disturbing other heavy bodies will be strong.

One can always stretch some math formulae for this tricky case, but Nature wont take much notice of them

Bye Fridger

[Malenfant]

Hm, well I figured something nasty like that might happen, I guess that treating them as pairs would be a reasonable simplification though (because I suppose there's no way I can figure out how it should really work because it's so complex... ) ?

That said, I did get the nested barycentre setup working...

(See also http://www.celestiaproject.net/forum/viewtopic.php?t=8431 for further discussion).

Code: Select all

Barycenter "Spica"
{
RA         201.29835230
Dec       -11.16124491    
Distance  262.05788
}
#-------------------------------
Barycenter "SpicaABCD"
{
   OrbitBarycenter "Spica"

   EllipticalOrbit {             
      Period          186989.398
      SemiMajorAxis   209   
      Eccentricity    0
      ArgOfPericenter 0
   }
}

"Spica E"
{
OrbitBarycenter "Spica"
SpectralType "K5V"
AbsMag  7.58 # Luminosity = 0.08 Sol
Radius 375983

   EllipticalOrbit {               
      Period          186989.398
      SemiMajorAxis   9791 
      ArgOfPericenter 180
   }
}
#-------------------------------
Barycenter "SpicaABC"
{
   OrbitBarycenter "SpicaABCD"

   EllipticalOrbit {             
      Period          47.80914
      SemiMajorAxis   5.7143   
      ArgOfPericenter 0
   }
}

"Spica D"
{
OrbitBarycenter "SpicaABCD"
SpectralType "B7V"
AbsMag  -1.14 # Luminosity = 250 Sol
Radius 1740662

   EllipticalOrbit {               
      Period          47.80914
      SemiMajorAxis   34.2857 
      ArgOfPericenter 180
   }
}
#-------------------------------
Barycenter "SpicaAB"
{
   OrbitBarycenter "SpicaABC"

   EllipticalOrbit {             
      Period          1.63299
      SemiMajorAxis   1   
      ArgOfPericenter 0
   }
}

"Spica C"
{
OrbitBarycenter "SpicaABC"
SpectralType "B5V"
AbsMag  -2.6 # Luminosity = 938 Sol
Radius 2088795
   EllipticalOrbit {               
      Period          1.63299
      SemiMajorAxis   3
      ArgOfPericenter 180
   }
}
#-----------------------

65474 "Spica A"
{
OrbitBarycenter "SpicaAB"
SpectralType "B1V"
AbsMag  -5.49 # Luminosity = 13400 Sol
Radius 5430867

   EllipticalOrbit {               
      Period          0.0098
      SemiMajorAxis   0.047 
      ArgOfPericenter 0
   }
}

"Spica B"
{
OrbitBarycenter "SpicaAB"
SpectralType "B4V"
AbsMag  -3.25 # Luminosity = 1710 Sol
Radius 2785060

   EllipticalOrbit {               
      Period          0.0098
      SemiMajorAxis   0.073 
      ArgOfPericenter 180
   }
}

#-------------------------------


[wcomer]

Fridger,

In general I would agree with your conclusion. However, for this particular case, assuming that there aren't any resonances, the cascaded distances between the bodies is sufficient to minimize the multibody interactions. Although the system is undeniably chaotic and thus nature will not follow these simple equations for very long. In the short term, Malenfant's strategy ought to be fine.

cheers,
Walton

[Malenfant]

I asked this on the thread on the development board, but I'll ask here too because it's a more astronomical question:

Putting a planet around Spica E (the K5 V star), I don't see any darkside illumination by the ABCD stars. Now, the appmag of the K5 V is about -26, and the appmag of the ABCD are -17.06, -14.82, -14.17, and -12.72, all crammed into 11 arcseconds of sky (about 1/3 the diameter of a full moon from Earth). The ABC triple are within about 1 arcsecond of eachother.

I'd guess that the ABC magnitudes would at least be cumulative somehow, wouldn't they? ie. they'd look like a single brighter star? Is there a way to calculate the cumulative magnitude? And should that be enough to light up the darkside (ie to be closer to -26)?

Or would the combined magnitudes probably be something like -17.5 because BCD are so much dimmer than A? Even if it is -17.5, that's still about 100 times brighter than the full moon - surely enough to visibly illuminate the darkside.

[chris]

I'd guess that the ABC magnitudes would at least be cumulative somehow, wouldn't they? ie. they'd look like a single brighter star? Is there a way to calculate the cumulative magnitude? And should that be enough to light up the darkside (ie to be closer to -26)?

The cumulative magnitude is easy to calculate--convert the magnitudes to light intensities, sum them, then convert back to magnitudes. The key thing is that the addition needs to be performed in a linear space like intensity rather than in the logarithmic magnitude scale. So you want to do this:

Msum = log base b(b^M0 + b^M1 + b^M2+ ...)

where b ~ 2.512 (the log base 5 of 100)

Actually, since the magnitude scale is backwards--an object with a lower magnitude is actually brighter--you should negate the powers of b:

Msum = -log base b(b^-M0 + b^-M1 + b^-M2 + ...)

What you'll find is that the -14.82, -14.17, and -12.72 magnitude stars don't contribute much.

Or would the combined magnitudes probably be something like -17.5 because BCD are so much dimmer than A? Even if it is -17.5, that's still about 100 times brighter than the full moon - surely enough to visibly illuminate the darkside.

It should barely be visible . . . Celestia modifies the contribution of each light source using by a gamma function to simulate the human eye's logarithmic response to light intensity. Pixel values generally range from 0 to 255; the gamma function is required to make light sources less than 1/255 as intense as the brightest light source contribute perceptibly. However, there's still a cutoff point where Celestia decides that a light source is too faint (relative to the brightest light source) to noticeably illuminate a planet. That value is currently set to 1/10000 the intensity of the brightest light source, or 10 magnitudes less. So, the light from the magnitude -17.5 star should just barely be visible. If you're using an LCD monitor--they don't perform as well as CRTs with very dark colors--you may not see anything.

--Chris

[Malenfant]

It should barely be visible . . . Celestia modifies the contribution of each light source using by a gamma function to simulate the human eye's logarithmic response to light intensity. Pixel values generally range from 0 to 255; the gamma function is required to make light sources less than 1/255 as intense as the brightest light source contribute perceptibly. However, there's still a cutoff point where Celestia decides that a light source is too faint (relative to the brightest light source) to noticeably illuminate a planet. That value is currently set to 1/10000 the intensity of the brightest light source, or 10 magnitudes less. So, the light from the magnitude -17.5 star should just barely be visible. If you're using an LCD monitor--they don't perform as well as CRTs with very dark colors--you may not see anything.

--Chris

Well, here's a problem with that as it stands:

Say I'm placed so that the planet is eclipsing the K5 V star, but the ABCD stars are on the opposite side of the planet so that they are fully illuminating the darkside. Here, I can't see the K5 V primary because it's behind the planet, so I should definitely be able to clearly see the illumination from the ABCD stars in this case.

I think I raised this issue before when you first implemented multiple lighting - namely that Celestia doesn't seem to account for the phase of the planet, it just assumes the brightness from all lightsources is the same regardless of how full or new the planet is from the observer's viewpoint. If you're really accounting for the response of the eye, then surely you have to make the dimmer illumination more noticeable in the cases where you're not seeing a lot of the brighter illumination.

Just like we can see earthshine on the moon when it's new or a thin crescent (ie mostly dark), but not when it's half-full, because the brightness of the lit side is drowning out the (reduced) illumination on the darkside.

Is there a way to account for this in Celestia? It'd make for a much more realistic response I think.