How can I determine the inclination degree? (visual binary)

[arcTomato]

Homework Statement

When I can observe the orbit of visual binary,How can I determine the inclination degree??

Relevant Equations

Keplar's low??

Problem Statement: When I can observe the orbit of visual binary,How can I determine the inclination degree??
Relevant Equations: Keplar's low??

But now, we know the distance to the binary(45.9pc), and viewing radius(1.2mas).

 

[lomidrevo]

and viewing radius(1.2mas).

what do you mean by viewing radius? The observed projections of semi-major axis?

Is this a real homework problem you have been given somewhere? It doesn't look like it is...

Generally, it may not be an easy task to determine the angle of inclination. As far as I know, the usual way to do it, is to compare the observed trajectories (orbit projected onto the plane of the sky) with projections of various ellipses. If the inclination is not zero, the focus of the observed orbit and the apparent position of the center of mass will not match, so seemingly the first Kepler's law will be violated. So the goal is to find such ellipse and inclination angle, for which the observed trajectories are consistent with the Kepler's law.

 

[arcTomato]

Hello @lomidrevo ,Thank you for helping, again and again!

what do you mean by viewing radius? The observed projections of semi-major axis?

Yes, sorry my english is bad...

If the inclination is not zero, the focus of the observed orbit and the apparent position of the center of mass will not match,

I'm not really sure this part.
So, If the inclination angle is ZERO, the focus equals the center of mass??

 

[lomidrevo]

So, If the inclination angle is ZERO, the focus equals the center of mass??

Yes, because if the inclination angle is zero, that means that orbital plane of the binary is perpendicular to your line-of-sight. In such case you see the orbit (ellipse) as it is described by first Kepler's law and the center of mass of the binary is located in one of the foci.

 

[lomidrevo]

Hello @lomidrevo ,Thank you for helping, again and again!

Com'on, its only third time
Although PF has many members, for some topics you will be meeting the same faces (or maybe I should say avatars) again and again. You will get used for that

And you are welcome, of course, but you don't have to thank each time. Members are always happy when they can help!

 

[arcTomato]

Yes, because if the inclination angle is zero, that means that orbital plane of the binary is perpendicular to your line-of-sight. In such case you see the orbit (ellipse) as it is described by first Kepler's law and the center of mass of the binary is located in one of the foci.

ok I have a new question.
Dose first Kepler's law describe "the center of mass of the binary is located in one of the foci" ?
I thought If I think of the sun and the earth, the center of mass of the binary is not the foci.But you said it's wrong, isn't it?

Com'on, its only third time
Although PF has many members, for some topics you will be meeting the same faces (or maybe I should say avatars) again and again. You will get used for that

And you are welcome, of course, but you don't have to thank each time. Members are always happy when they can help!

ok but I just wanted to tell you how much I am grateful!
I'm studying about astrophysics ,so I'll often ask a question where I don't know.

 

[lomidrevo]

Dose first Kepler's law describe "the center of mass of the binary is located in one of the foci" ?
I thought If I think of the sun and the earth, the center of mass of the binary is not the foci.

Well yes, the "original" law defined by Kepler said that the Sun is in the focus. But revisiting the law using Newtonian mechanics and his Gravitational law, you will see that it is not Sun, but the center of mass of the system that sits in the common focus.

We cannot blame Kepler for this "small" incorrectness, because the observations were not precise enough. And as Sun contains more than 99% of mass of the whole solar system, it is not surprising.

 

[lomidrevo]

I'm studying about astrophysics ,so I'll often ask a question where I don't know.

OK, so very soon it will be quite contrarily: you will answer my questions I am just an enthusiastic amateur.

 

[arcTomato]

Well yes, the "original" law defined by Kepler said that the Sun is in the focus. But revisiting the law using Newtonian mechanics and his Gravitational law, you will see that it is not Sun, but the center of mass of the system that sits in the common focus.

So,There is a little bit distance from the real center of mass to the location of the sun, But can we ignore it??right??

If the inclination is not zero, the focus of the observed orbit and the apparent position of the center of mass will not match,

Really?I can't imagine this part.
I thought that Line-of-sight direction of the component of the center of mass is the same.

So the goal is to find such ellipse and inclination angle, for which the observed trajectories are consistent with the Kepler's law.

Can I calculate this on hand?? How??

OK, so very soon it will be quite contrarily: you will answer my questions I am just an enthusiastic amateur.

Wow,Amateur! Cool!
I have to study so hard

 

[lomidrevo]

So,There is a little bit distance from the real center of mass to the location of the sun

The common center of mass (barycenter) is not located in the center of Sun.
https://en.wikipedia.org/wiki/Barycenter

But can we ignore it??

I would say, it depends on what kind of task you are going to perform.

Really?I can't imagine this part.

See figure 4.3 https://www.ast.cam.ac.uk/~pettini/STARS/Lecture04.pdf.

Can I calculate this on hand?? How??

You mean to find the matching ellipse and angle of inclination? This is a task for a computer I would say.

Also have a look at equation (4.4) in the same link. Normally, you are trying to guess the total and individual masses of the binary. But if these are know to you already, you can indeed calculate the angle of inclination by hand.

 

[arcTomato]

I got it now! This link is so useful!