#### arcTomato

Summary
Hello

I have no idea why underline part of my pic says like that.
"Radio observations show that the component of this motion perpendicular to the Galactic plane is at most 0.4 ± 0.9 km s−1"
So What???? Why this sentence shows the mass??
("The radio emitting source(underline part)" = Sgr A*???right??)

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#### phyzguy

The star S2 and the radio source SgrA* are orbiting about their common center of mass. The ratio of their orbital velocities is inversely proportional to the ratio of their masses. In other words, the more massive one moves slower. We can see the motion of S2 so we know its orbital velocity about SgrA*. The radio observations put an upper limit on the speed of the orbital motion of SgrA*. We know (roughly) the mass of the star S2, so from the ratio of the orbital velocities we can put a lower limit on the mass of SgrA*. Does this make sense?

#### arcTomato

We can see the motion of S2 so we know its orbital velocity about SgrA*.
How can I calcualte this??Law of angular momentum conservation and Law of energy conservation????

And Why dose orbital velocity connect the motion perpendicular to the Galactic plane ??

#### phyzguy

How dose it calcualte??Law of angular momentum conservation and Law of energy conservation????
Are you asking how we calculate the velocity of S2? You can see its orbit, so you know the angular size of the orbit. You know how far away it is, so you can calculate the actual size of the orbit in km. You can see from the times how long it takes to orbit (about 20 years). So its a simple matter to calculate its orbital velocity. Is that your question?

#### arcTomato

??
I can't understand yet
The velocity of S2 is not constant???right??

and how we know the mass of S2???

#### phyzguy

??
I can't understand yet
The velocity of S2 is not constant???right??

and how we know the mass of S2???
I'm being approximate and trying to give you the idea of how it's done. Remember this is just an order-of-magnitude calculation, since the measured velocity of SgrA* was 0.4+/-0.9 km/s, and is very uncertain. If the orbit of S2 were circular, then the velocity would be constant. It's actually an ellipse, so the velocity varies, but from Kepler's second law this is easy to calculate. As for the mass, S2 is a star. We know the masses of stars and how the masses depend on how bright they are. We know how far away it is and we know how bright it is, so we know about how massive it is.

#### arcTomato

I'm being approximate and trying to give you the idea of how it's done. Remember this is just an order-of-magnitude calculation, since the measured velocity of SgrA* was 0.4+/-0.9 km/s, and is very uncertain. If the orbit of S2 were circular, then the velocity would be constant. It's actually an ellipse, so the velocity varies, but from Kepler's second law this is easy to calculate. As for the mass, S2 is a star. We know the masses of stars and how the masses depend on how bright they are. We know how far away it is and we know how bright it is, so we know about how massive it is.
Ok I think I can do this now, thanks for your kindness!

#### arcTomato

P.S.
Finally I can understand almost this sentence!
thank you!
But I do not know only one place that
The ratio of their orbital velocities is inversely proportional to the ratio of their masses. In other words, the more massive one moves slower.
.
How can I lead this ratio?

EDIT: Someone who can teach me , PLEASE!

Last edited:

#### lomidrevo

How can I lead this ratio?
Let's pretend that the orbital eccentricity is low, so we approximate the orbit as circular. Then, what is the relationship between the tangent (orbital) speed $v$ of an object, the radius of its orbit (semi-major axis) $a$ and its orbital period $P$?
Once you figure this out for both objects and combine it with the fact that they are orbiting about their common center of mass, you will find what you are looking for.
Hint: Obviously, the more massive object must be closer to the center of mass of the binary system.

#### arcTomato

Wow lomidrevo it's you!, thank you again!!

what is the relationship between the tangent (orbital) speed $v$ of an object, the radius of its orbit (semi-major axis) $a$ and its orbital period $P$?
I think $P=2πa/v$,and

Is it right??

but this orbit is ellipse, Why this ratio still works???

#### phyzguy

but this orbit is ellipse, Why this ratio still works???
This ratio is just saying that in the center of mass of the system the total momentum is zero, so p(sgrA*) = -p(S2).

#### arcTomato

This ratio is just saying that in the center of mass of the system the total momentum is zero, so p(sgrA*) = -p(S2).
y I realize it now.
Thank you!

#### lomidrevo

Is it right??
perhaps it could be reduced to fewer steps, but the result is ok. E.g. the employment of the third Kepler's law would not be needed. Remember you want to find a ratio of velocities, so quantities like $P$ and $a$ cancel out.
Anyway, the argument provided by @phyzguy is far more elegant and generalizes to eccentric orbits as well.

#### arcTomato

OK, Thank you so much @lomidrevo and @phyzguy.
Your explanation are so easy to understand.

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