Alternative Splash image; Double Star orbits

[t00fri]

Further thought:

what I could easily do for each binary in the 6th catalog via Perl, Kepler's 3rd law above and the respective colors from 'stars.txt' (Hipparcos data), would be a plot of the Primary color versus m1+m2, the sum of masses of the binary system.

If we see some color--mass correlation, we could exploit it...

Bye Fridger

[granthutchison]

My only additional thought when I was looking at this was that, since the 6th catalog lists the WDS coordinates for every pair, it would be possible to extract secondary spectral types when these are listed in WDS. Unfortunately, it appeared that barely one in 50 WDS entries contained a second spectral type, and that was presented in no standard format I could discern.

Grant

[Evil Dr Ganymede]

If we see some color--mass correlation, we could exploit it...

There's obviously a colour-mass correlation.

Since you seem to be ignoring me unless I'm "useful" , here you go - this is the ZAMS for stars with z=0.001 and for z=0.020 from the Geneva Evolution files. (my spectral types may be incorrect because I can't find a single agreed-on list of spectral type vs temperature, but the temperatures are right as read from the files).

I can't get the formatting straight here, but if you copy and paste into a text file it should be more obviousl But again, this is only the Zero Age Main Sequence for these stars, and doesn't say anything about what they evolve into.

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Age      M   Log T   Log L   T/K   Ls   Ts   Rs   R/AU   Spec Type
lots      0.09   3.430   -4.6   2692   0.00003   2691.53   0.1015   0.0005   M5 VI
lots      0.1   3.504   -3.5   3192   0.00032   3191.54   0.11   0.0005   M2 VI
lots      0.2   3.557   -2.20   3606   0.00631   3605.79   0.195   0.0009   K9 VI
lots      0.3   3.585   -1.78   3846   0.017   3845.92   0.28   0.0013   K8 VI
1.62E+09   0.4   3.619   -1.484   4159   0.033   4159.11   0.365   0.0017   K6 VI
7.94E+08   0.5   3.641   -1.2   4375   0.063   4375.22   0.45   0.0021   K4 VI
4.53E+08   0.6   3.686   -0.863   4853   0.137   4852.89   0.535   0.0025   K1 VI
2.54E+08   0.7   3.737   -0.546   5458   0.28   5457.58   0.62   0.0029   G5 VI
9.00E+07   0.8   3.77   -0.3   5888   0.50   5888.44   0.705   0.0033   G1 VI
4.00E+07   0.9   3.801   -0.062   6324   0.87   6324.12   0.79   0.0037   F8 VI
4.00E+07   1   3.828   0.153   6730   1.42   6729.77   0.875   0.0041   F5 VI
4.00E+07   1.25   3.91   0.592   8128   3.91   8128.31   1.00   0.0046   A8 VI
8.51E+07   1.5   3.986   0.932   9683   8.55   9682.78   1.04   0.0048   A4 VI
8.68E+07   1.7   4.034   1.157   10814   14.35   10814.34   1.08   0.0050   A0 VI
1.90E+07   2   4.086   1.411   12190   25.76   12189.90   1.14   0.0053   B9 VI
1.48E+07   2.5   4.151   1.773   14158   59.29   14157.94   1.28   0.0059   B8 VI
1.19E+06   3   4.207   2.057   16106   114.02   16106.46   1.37   0.0064   B6 VI
1.00E+06   4   4.276   2.492   18880   310.46   18879.91   1.65   0.0077   B4 VI
1.17E+06   5   4.325   2.817   21135   656.15   21134.89   1.91   0.0089   B3 VI
                           
                           
                           
                           
Age      M   Log T   Log L   T/K   Ls   Ts   Rs   R/AU   Spec Type
(BD)      0.07   3.340   -5.00   2188   0.00001   2187.76   0.1095   0.0005   M8 V
Proxima      0.1   3.430   -4.30   2692   0.00005   2691.53   0.135   0.0006   M5 V
Barnard      0.2   3.482   -3.40   3034   0.0004   3033.89   0.22   0.0010   M4 V
Gliese 876   0.3   3.544   -2.00   3499   0.0100   3499.45   0.305   0.0014   M0 V
4.30E+08   0.4   3.572   -1.64   3733   0.0229   3732.50   0.39   0.0018   K8 V
8.13E+08   0.5   3.595   -1.37   3936   0.0427   3935.50   0.475   0.0022   K7 V
5.28E+08   0.6   3.623   -1.09   4198   0.0813   4197.59   0.56   0.0026   K5 V
3.36E+08   0.7   3.654   -0.821   4508   0.1510   4508.17   0.645   0.0030   K3 V
1.27E+09   0.8   3.692   -0.576   4920   0.27   4920.40   0.73   0.0034   K1 V
1.00E+08   0.9   3.724   -0.352   5297   0.44   5296.63   0.815   0.0038   G7 V
5.00E+07   1   3.755   -0.132   5689   0.74   5688.53   0.9   0.0042   G3 V
3.30E+08   1.25   3.811   0.367   6471   2.33   6471.43   1.22   0.0056   F7 V
6.00E+07   1.5   3.862   0.711   7278   5.14   7277.80   1.43   0.0066   F1 V
8.51E+07   1.7   3.911   0.949   8147   8.89   8147.04   1.50   0.0070   A8 V
1.10E+07   2   3.958   1.209   9078   16.18   9078.21   1.63   0.0076   A5 V
7.00E+06   2.5   4.031   1.6   10740   39.81   10739.89   1.83   0.0085   A0 V
2.36E+06   3   4.088   1.909   12246   81.10   12246.16   2.00   0.0093   B9 V
1.00E+06   4   4.173   2.385   14894   242.66   14893.61   2.34   0.0109   B7 V
1.00E+06   5   4.235   2.74   17179   549.54   17179.08   2.65   0.0123   B6.5 V


(the star's mass is in the second column). The "age" is just the age (in years) at which the grid starts, usually assumed to be when the star forms - if it's a few billion years then you can assume that the star is like that up til that time). Ls, and Rs are luminosity in Sols, and Radius in Sols.

[t00fri]

My only additional thought when I was looking at this was that, since the 6th catalog lists the WDS coordinates for every pair, it would be possible to extract secondary spectral types when these are listed in WDS. Unfortunately, it appeared that barely one in 50 WDS entries contained a second spectral type, and that was presented in no standard format I could discern.

Grant

Grant,

funny coincidence...

just at this same time I happened to investigate the same issue . For my modem the download of the WDS takes too long, but I still have the XEphem converted results from the WDS. It seemed to me from there that there should be quite many systems with both color classes specified. But you certainly had a more direct view.

Bye Fridger

[t00fri]

If we see some color--mass correlation, we could exploit it...

There's obviously a colour-mass correlation.

..
- this is the ZAMS for stars with z=0.001 and for z=0.020 from the Geneva Evolution files. (my spectral types may be incorrect because I can't find a single agreed-on list of spectral type vs temperature, but the temperatures are right as read from the files).

I can't get the formatting straight here, but if you copy and paste into a text file it should be more obviousl But again, this is only the Zero Age Main Sequence for these stars, and doesn't say anything about what they evolve into.

....

(the star's mass is in the second column). The "age" is just the age (in years) at which the grid starts, usually assumed to be when the star forms - if it's a few billion years then you can assume that the star is like that up til that time). Ls, and Rs are luminosity in Sols, and Radius in Sols.

Evil Dr.

This looks like worth a try.

I should just understand a little better. Can you point me to the original file and remind me what the Geneva Evolution files are precisely? Now this is only a small sample of stars ; how exactly was it selected apart from the z-window? What is of particular interest is the spread in the temperature-mass correlation, in order to arrive at a feel for the uncertainties involved in this method.

Once we have some reasonably sensible correlation of
color/temperature versus mass, we could inject the estimate of the primary mass from the quoted color information in the Hipparcos data (stars.txt) for our binaries and try to get hold of the seconday mass from knowing m1+m2.

In some prominent cases one could cross check the results against the published individual masses.

Very interesting...

Bye Fridger

[Evil Dr Ganymede]

Here are the grids of which I speak:
http://obswww.unige.ch/~mowlavi/evol/stev_database.html

Click on the links there and you'll get to loads of evolutionary tables for different masses and metallicities.

The masses from 0.8 Sols and up are provided in the tables there... I can't recall where I found the ones below that, I think I got those from data on stars of those masses and generalising somewhat.

The uncertainties are probably huge, to be honest. This is all what theory suggests, not what is actually out there (which is complicated by all sorts of things like compositional differences and just being randomly awkward) - so you should probably think of them more as guidelines.

[t00fri]

Grant and friends,


... exploiting a bit my vacations, I spent several hours today reading in various relevant books I happen to have in my bookshelf. I think I understood a number of things now that bugged me previously:

Here we go:

One typically may consider 3 coordinate systems for our problem of binary orbits:

i) Observer at rest (i.e. at the coordinate origin). In this frame the barycenter moves linearly with time

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      x_b(t) = a  t + b ;   (x_b,  x_1, x_2, a,b are all vectors)
      x_1(t) = x_b(t) -  m2/(m1+m2) x(t)
      x_2(t) = x_b(t) + m1/(m1+m2) x(t)
      x(t)     = x_2(t) - x_1(t)
     

ii) Barycenter at rest, i.e. x_b(t) = 0, observer is comoving. Here we have

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      x_b(t) = 0
      x_1(t) = - m2/(m1+m2) x(t)
      x_2(t) =   m1/(m1+m2) x(t)
      x(t)     =   x_2 (t)  - x_1(t)
      |x_1/x_2| = m2/m1
     

iii) Primary at rest, the system that is used in all
practical observations! (observer comoving)

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       x_1(t) = 0
       x_2(t) = x(t)
       x_b(t) = m2/(m1+m2) x(t)
     

It was my impression that some of the previous discussions mixed these systems implicitly or explicitly up.

Clearly all quoted orbital elements in catalogs for visual binaries refer to system iii) where the primary is at rest.

Next let me remember,

Why we need exactly 6 orbital elements to characterize the secondarie's orbit in 3d space?
From what observations are these 6 parameters derived in practice!?

(answer):

a) simplest and approach
---------------------------------------
In the limit that only the gravitational force of the primary is considered ( m2/m1 -> 0),
the secondary's orbit arises --after reduction of the 2-body problem to an effective 1-body problem --in terms of the difference vector x(t) as solution of the equation of motion.

The latter is a second order differential equation that we all know and love . To make this initial value problem unique, we have to prescribe at initial epoch t0 both x(t0) and dx(t0)/d t =v(t0), the initial velocity of the secondary.

In 3d space the initial position x(t0) and velocity v(t0) of the secondary in frame iii) precisely comprise 6 independent parameters, directly accessible through binary star observations.

It is a simple but instructive exercise of celestial mechanics to relate these 6 initial values to the standard 6 orbital elements (P,e,i,Omega,w, Periheltime) in a convenient frame

The assumption of a non-vanishing m2/m1 ratio will lead to the so-called "osculating elements" for which m2/m1 needs to be known. Let me turn to this case next:

=========>

b) In reality a nonvanishing secondary mass causes also a gravitational field that feeds back onto the primary. Yet one may take account of this by rescaling the initial data simply as follows:

replace the gravitational constant

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               G^2  -> G^2*(1 + m2/m1)


in the equations of motions and modify the initial data by

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x(t0)
v(t0)/G -> v(t0)/(G*sqrt(1+m2/m1))


Obviously, the new determination of the orbital elements
now requires also as 7th parameter m2/m1 as input. I have done all those calculations today.

The net effect is that in the orbital elements i,Omega,w
the m2/m1 cancels out, while in the conic section (ellipse) parameters the m2/m1 dependence appears in consistency with the 3rd Kepler law and the barycenter rule.

I hope this clears up some issues.

Bye Fridger

[t00fri]

""

[granthutchison]

I'm not seeing it, I'm afraid.
The derived conic section surely applies independently of the position of the barycentre, with only a necessary change in the radius of gyration of each body. The body-centred and barycentric elements are otherwise identical.

Grant

[granthutchison]

I'm not seeing it, I'm afraid.
The derived conic section surely applies independently of the position of the barycentre, with only a necessary change in the radius of gyration of each body. The body-centred and barycentric elements are otherwise identical.

Are we using different assumptions on how these orbits were derived?
I'm assuming that a Keplerian fit is made to the observed change in angular separation and position angle against time. The shape and orientation of an orbit derived in this way is independent of the barycentre position.
Are you starting from the same place?

Grant

[t00fri]

I'm not seeing it, I'm afraid.
The derived conic section surely applies independently of the position of the barycentre, with only a necessary change in the radius of gyration of each body. The body-centred and barycentric elements are otherwise identical.

Grant

Grant,

too bad, you are (mostly?) right, of course.

I was too inpatient , as to my conclusions:

since I calculated in units m1=1, I had overlooked another m-dependence after (too quickly ) rescaling back to general m1. So after correction of this blunder, we seem to be back at the old story.

So I'll edit my above post accordingly, in order to avoid further confusion.

Bye Fridger

[t00fri]

I'm not seeing it, I'm afraid.
The derived conic section surely applies independently of the position of the barycentre, with only a necessary change in the radius of gyration of each body. The body-centred and barycentric elements are otherwise identical.

Are we using different assumptions on how these orbits were derived?
I'm assuming that a Keplerian fit is made to the observed change in angular separation and position angle against time. The shape and orientation of an orbit derived in this way is independent of the barycentre position.
Are you starting from the same place?

Grant

I am not really sure. I tried to minimize assumptions and started right from the equations of motion for two masses m1,m2 in a central potential together with the initial values x(t0), v(t0). The general conic section solution

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x(phi) = p/(1+e*cos(phi))


after splitting off the movement of the barycenter, is obvious.

The less trivial exercise is to use a quite elegant method from my textbook on celestial mechanics to calculate all orbital elements in an arbitrary frame in terms of the initial data for the secondary x(t0), v(t0) in the primary's rest frame. So I have explicit formulae for all orbital parameters and nothing is fitted at that point.

I am still not 100 % sure whether I get entirely the old story...continue working on it...

Bye Fridger

[granthutchison]

too bad, you are (mostly?) right, of course.

In my life, being (mostly?) right is a good thing. One snatches what joy one can.

Are we using different assumptions on how these orbits were derived?
I'm assuming that a Keplerian fit is made to the observed change in angular separation and position angle against time. The shape and orientation of an orbit derived in this way is independent of the barycentre position.
Are you starting from the same place?

I am not really sure.

Okay. I wondered if you were working from (say) proper motions and separations, which would (I think, off the top of my head) produce very different orbits depending on the assumed motion and position of the barycentre.
(Hmmm. Or would the observed separation and proper motions (if measured exactly!) be compatible with only a single barycentre?)

Grant

[Evil Dr Ganymede]

Why we need exactly 6 orbital elements to characterize the secondarie's orbit in 3d space?
From what observations are these 6 parameters derived in practice!?

My Solar System Dynamics book isn't here at the mo, but IIRC you need the following to fully describe the orbit in 3D space:

semimajor axis - the distance from the primary.
eccentricity - how non-circular the orbit is.
inclination - angle between a reference plane and the orbital plane.
longitude of pericentre - angle between the pericentre of the orbit and a reference direction.
longitude of ascending node - angle between the reference line in the reference plane and the line where the orbital plane intersects the reference plane (the "line of nodes").
Argument of pericentre - angle within orbital plane between line of nodes and pericentre of orbit.


Also:
The True Anomaly is the angle between the current position of the object in its orbit, and the pericentre of the orbit.

I notice that Celestia uses the Mean Anomaly (M) instead of the True Anomaly. The Mean Anomaly is a rather weird parameter that is pretty damn near impossible to visualise. M is defined as n(T-t) where n is the mean motion, and T-t is the time elapsed since pericentre passage. While it is an angle, it has no simple geometric interpretation.

[Evil Dr Ganymede]

Why do you need to worry about reference frames here? Can you not just say the barycentre is fixed in space and everything orbits around it? Isn't that basically the centre of mass of the system?

[t00fri]

Why we need exactly 6 orbital elements to characterize the secondarie's orbit in 3d space?
From what observations are these 6 parameters derived in practice!?

My Solar System Dynamics book isn't here at the mo, but IIRC you need the following to fully describe the orbit in 3D space:

semimajor axis - the distance from the primary.
eccentricity - how non-circular the orbit is.
inclination - angle between a reference plane and the orbital plane.
longitude of pericentre - angle between the pericentre of the orbit and a reference direction.
longitude of ascending node - angle between the reference line in the reference plane and the line where the orbital plane intersects the reference plane (the "line of nodes").
Argument of pericentre - angle within orbital plane between line of nodes and pericentre of orbit.


Also:
The True Anomaly is the angle between the current position of the object in its orbit, and the pericentre of the orbit.

I notice that Celestia uses the Mean Anomaly (M) instead of the True Anomaly. The Mean Anomaly is a rather weird parameter that is pretty damn near impossible to visualise. M is defined as n(T-t) where n is the mean motion, and T-t is the time elapsed since pericentre passage. While it is an angle, it has no simple geometric interpretation.

There is clearly a multitude of different parametrizations one may chose for describing the orbit in space. I am certain that in the approximations I was discussing above there are only 6 independent parameters, corresponding to 3 in x(t0) and 3 in v(t0). If you take nonvanishing m2/m1, there is one more.
Should be obvious? Once you consider the nonvanishing m2/m1 case, you have 7 as I stated above. So your textbook presumably refers to the more general case.
But I think I stated the scenarios clearly above.

Bye Fridger

[t00fri]

Why do you need to worry about reference frames here? Can you not just say the barycentre is fixed in space and everything orbits around it? Isn't that basically the centre of mass of the system?

I think I tried to explain this, too, above. Your statement is only correct in a particular coordinate frame!

In my notation, in frame ii). In the frames i) (observer's rest frame) and iii) (primary's rest frame), the barycenter moves linearly with time. It's the solution of the barycentric equation of motion:

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(m1+m2)* d^2 x_b(t)/dt =0


Since the baricenter moves force free, the rhs is zero!
Upon integration you get the linear movement with time,

x_b(t) = a t + b

with a,b being integration constants. You may chose a special frame with a=b=0. This corresponds to frame ii).

You do not seem to read my posts well before answering. Sorry, if my English makes you suffer...

The statements about the movement of the barycenter are frame dependent. I just can't help that

The physics is, of course independent of the choice of frame, hence we may choose the most convenient one for the problem at hand.

Bye Fridger

[granthutchison]

semimajor axis - the distance from the primary.
eccentricity - how non-circular the orbit is.
inclination - angle between a reference plane and the orbital plane.
longitude of pericentre - angle between the pericentre of the orbit and a reference direction.
longitude of ascending node - angle between the reference line in the reference plane and the line where the orbital plane intersects the reference plane (the "line of nodes").
Argument of pericentre - angle within orbital plane between line of nodes and pericentre of orbit.

Also:
The True Anomaly is the angle between the current position of the object in its orbit, and the pericentre of the orbit.

You can get by with either the longitude of the pericentre or the argument of the pericentre, since longitude of pericentre = ascending node + argument of pericentre. The remaining five plus some measure of the anomaly gives you Fridger's six parameters to define an orbit in space.

I'm agreeing with Fridger that his seventh parameter (some measure of the mass ratio) does alter the orbital parameters. But I'm sure all it does is allow us to split the original elements into two complementary sets, consisting of two orbits with the same shape and plane as the original, their semimajor axes proportioned according to the mass ratio, and their pericentre arguments opposed - that is, the barycentric orbits of the two components, rather than the body-centred orbit of one component around the other.

Grant

[t00fri]

What is interesting is that the 6th catalog also uses 7 parameters:

Besides the ones that are obvious, and in agreement with what I discussed above,

--semi-major axis (a)
--eccentricity (e)

--inclination (i)
--node (Omega)
--longitude of periastron (w)
--time of periastron passage (T)

they quote as 7th parameter

--the period (P) in years.

Now I wondered what it corresponds to in addition to what I was discussing above.

Clearly the conic section (elliptic orbit) is uniquely specified by the semi-major axis a and the eccentricity e.

The period of the orbit measures according to the 3rd Kepler law for given semi-major axis a, the sum m1+m2 of the masses. Hence, these set the scale for the time-duration of the orbit, but this parameter does not contribute to a specification of it's shape in space.

Any other ideas?

Bye Fridger

[Evil Dr Ganymede]

You do not seem to read my posts well before answering. Sorry, if my English makes you suffer...

The english isn't making me suffer, its the maths... (well, maybe it's both. The fact you're not explaining things much in non-mathematical terms probably isn't helping)

Back up a bit though. If you can have a frame of reference where everything in the system orbits the barycentre, why does it have to be moving at all? Wouldn't the barycentre act as if it's an object that has the mass of the two stars, and everything else would follow an elliptical orbit around it?