B Does Sun cross planes of all binary stars?

[snorkack]

The radiant of peculiar motion of any star relative to Sun is antipodal to its antiradiant. Likewise the radiant of peculiar motion of Sun relative to any star is antipodal to its antiradiant. Correct?
But any bound orbit, being an ellipse, is a great circle. Therefore any pair of antipodes is on opposite hemispheres of any great circle and any line connecting any pair of antipodes crosses any great circle.
Does it follow that the proper motion of Sun will always cross the plane of any binary star?
How are the epoch and angular speed of such crossing derived from the astrometric data commonly reported?

 

[.Scott]

Does it follow that the proper motion of Sun will always cross the plane of any binary star?

I'm not sure which plane you are talking about - most likely the plane that both binary stars follow.
But it probably doesn't matter which plane you mean. In general, our Sun may not complete an orbit around the binary stars, so it may not cross their plane.
So, no.

 

[snorkack]

I'm not sure which plane you are talking about - most likely the plane that both binary stars follow.
But it probably doesn't matter which plane you mean. In general, our Sun may not complete an orbit around the binary stars, so it may not cross their plane.
So, no.

But Sun in general completes a half-orbit - a straight line infinity to infinity.
And a straight line infinity to infinity is half of a great circle.
Since any two great circles cross at two points, any whole great circle and half great circle should cross at one point?

 

[Ibix]

The Sun's path is not necessarily a straight line and could be curved away from the binary pair. Nor is its lifetime infinite, so there's only a finite line segment to work with.

An obvious example would be a binary pair in another galaxy with their orbital plane more or less perpendicular to the line from them to the Sun. It could easily stay away from the binaries' orbital plane for a few billion years.

Note, however, that I do not know if orbital planes are stable over that long a time period. You may find that the binary system's orbital plane sweeps over us even if it wasn't initially anywhere near us.

 

[.Scott]

But Sun in general completes a half-orbit - a straight line infinity to infinity.
And a straight line infinity to infinity is half of a great circle.
Since any two great circles cross at two points, any whole great circle and half great circle should cross at one point?

The Sun may be over the escape velocity of the binary.

 

[snorkack]

The Sun may be over the escape velocity of the binary.

Which is why Sun completes less than a whole great circle.
If Sun were at exactly escape velocity then a parabola completes almost whole great circle, with no opening angle.
If Sun were at over escape velocity, but at some finite velocity, then Sun would complete more than a half of a great circle, because a hyperbola has a definite opening angle.
If the binary had zero escape velocity then the Sun would travel in straight line unperturbed by the binary and therefore complete half of a great circle.

 

[sophiecentaur]

Does it follow that the proper motion of Sun will always cross the plane of any binary star?

Are you wondering how binary stars can always be detected by occlusion? The sun could be anywhere relative to the orbital plane of the binary pair and the probability of the three stars being aligned could be very small. You'd have to detect at least two occlusions to have a clue; it could take centuries.

 

[snorkack]

Are you wondering how binary stars can always be detected by occlusion? The sun could be anywhere relative to the orbital plane of the binary pair and the probability of the three stars being aligned could be very small. You'd have to detect at least two occlusions to have a clue; it could take centuries.

No. It seems to me that from first principles of geometry, any star should by its proper motion cross any great circle at some epoch - even if it is some distant future or past one - and therefore Sun should be on its way across the orbital plane of any binary.
Anybody having a redshift would have more than half of its orbit - itself half of a great circle as pointed out above - in the past, and less than half of its orbit in the future.
I understand that for stars in distant galaxies, the epochs of their closest approach to Sun tend to cluster around 14 Gyr in the past, but with significant scatter, such as stars in Andromeda nebula that have epoch of closest approach in future.

 

[.Scott]

If the binary is in the Milky Way galaxy, then the plane shared by the binary will scan through most stars in our galaxy on each of it's Galactic years - and the stars that it misses will necessarily be caught within the galactic year of each of those stars. For the Earth, a galactic year is about 225 Million years - a period of time that is well within the life span of most stars and binaries.

On the other hand, if the binary is not in our galaxy, then our Sun may have been formed already travelling away from that binary at well over that binary's escape velocity. In that case, there is no guarantee that the sun will ever cross through the binary's plane. It is even possible that no part of the Milky Way galaxy will ever pass though such a binary's plane.

 

[sophiecentaur]

No. It seems to me that from first principles of geometry,

It's true that the two orbital planes must intersect but detecting an orbit would need to be by using frequency shift and not occlusion unless we were very lucky.

 

[snorkack]

On the other hand, if the binary is not in our galaxy, then our Sun may have been formed already travelling away from that binary at well over that binary's escape velocity. In that case, there is no guarantee that the sun will ever cross through the binary's plane.

Generally there is no guarantee that "ever will". A straight line orbit being exactly half of a great circle is guaranteed to cross any other great circle in past or future but is not guaranteed to cross that great circle specifically in the future. It may cross the orbit, for instance, before the Sun, the orbit or indeed the world formed.

 

[sophiecentaur]

even if it is some distant future or past one

If you include the linear motions of the Sun and the binary then you have two helices which may only intersect one or two times before being lost for ever (only one opportunity for an occlusion).

 

[snorkack]

If you include the linear motions of the Sun and the binary then you have two helices which may only intersect one or two times before being lost for ever (only one opportunity for an occlusion).

Or zero times. A star with a proper motion crosses an orbit at a definite epoch with a definite angular velocity. Depending on the said angular velocity and the width of the eclipse strip, the duration of the period in which eclipses happen may be shorter than the orbital period, in which case one eclipse may or may not fall in that period.

 

[sophiecentaur]

the period in which eclipses happen may be shorter than the orbital period, in which case one eclipse may or may not fall in that period.

Intuitively (not a good approach, I know) you'd think that most binaries would be missed. But as techniques improve we should see more - a bit like the exoplanet hunt.

 

[snorkack]

Intuitively (not a good approach, I know) you'd think that most binaries would be missed. But as techniques improve we should see more - a bit like the exoplanet hunt.

A great majority of all binaries have the epoch of their closest approach clustered around 14 Gyr in the past. Both sides of it. Duh.
How are the distances of their closest approach distributed?