BaryCenter

            In astronomy, the barycenter is the center of mass of two more bodies, that orbit one another & is the point at which the bodies are in orbit. planets that are in our solar system are not orbiting our sun they are orbiting the barycenter of the sun & whole planets in our solar system.

BaryCenter Of Sun

            If one of two orbiting bodies is much more massive than the other & the bodies are relatively closed to one another, the barycenter will typically be located within the more massive object. in this case, rather than the two more bodies appearing to be orbit a point. between them, the less massive body will appear to orbit about the more massive body. while the more massive body might be observed to wobble slightly. this is the case of the earth-moon system, in which the barycenter is located on average like 4671 km, from earth's center,75% of earth radius of 6378 Km when the two bodies of similar masses, the barycenter will generally be located between them & both bodies will orbit around it. this is the case for pluto & Charon. one of pluto natural satellites as well as many binary asteroids & binary stars, when less massive object far away, the barycenter can be located outside the more massive object. this is the case of Jupiter & the sun; despite the sun being 1000 folds more massive than Jupiter, their barycenter is slightly outside the sun due to the relatively large distance between them.

            In astronomy, barycenter coordinates are none rotating coordinated with the origin at the barycenter of two or more bodies. the international celestial reference system is a barycentric coordinate system centered on the solar system's barycenter.

Two Body Problem :

             The barycenter is one of the foci of the elliptical orbit of each body id "a" is a distance between the center of two bodies, "r1" is the semi-major axis of the secondary's orbit. when the barycenter is located within a more massive body that body will appear to "wobble" rather than to follow a discernible orbit. in a similar simple two-body case, the distance from the center of the primary to the barycenter "r1" is given by,    

                             m2                      a

          r1=a.    -----------    =      ---------------
                        m1+m2        1 + m1/m2

Where, 

'r1' , is the distance from body 1 to the barycenter

'a' , is the center between the two bodies.

'm1' & 'm2' are the masses of the primary & secondary bodies respectively.

Primary - secondary examples =

The terms "primary" & "secondary" are used to distinguish between involved participants, with the large being the primary & the smaller, being the secondary. 

m1 is the mass of the primary.

m2 is the mass of the secondary.

a, is the average distance between two bodies from their center

r1, is the distance from the center of the primary to barycenter.

R1 is the radius of the primary 

R2 is the radius of the secondary

r1/R1, a value less than 1 means the barycenter lies insight the primary

For eg, Consider finding the barry center of the earth & the moon

rE=6378.1 Km =radius of earth

rM=1737.1 Km=radius of moon

ME=5.972*10^24 Kg = mass of the earth

MM=7.34767309*10^22 Kg = mass of the moon

d=384400 +- Km = distance between the earth & moon.

a=392515.2 +-Km= distance between the earth & moon from their centers 

            i.e  CE-CM

           where ;as we known, 

                           a                     a

r1 =               -------------  =  -----------

                        1+m1/m2      1+ME/MM

         384300*10^3

r1 = ----------------------

                5.972*10^27

         1 +  -------------------------

                7.34767309*10^25

         r1 = 4670.78267*10^3

            r1 = 4670.78627 Km

This is the BaryCenter of the earth & moon from the center of the earth.

r2 = a - r1 =  (384300*10^3 + 6378.1*10^3 + 1737.1*10^3) - 4670.782627*10^3

r2 = 387844.42 Km

This is the BaryCenter of the earth & mon from the center of the moon

            Some time BaryCenter is ain between the body, like earth & moon, sun & earth,

so we define that by the ratio of the distance of the center of the primary to the BaryCenter to the radius of the primary.

ie eg. about the earth & moon.

    r1                4670.462627*10^3

-----------  =  --------------------------    =   0.73231

   R1                6378.1*10^3

It means the BaryCenter of the (earth-moon) is inside the earth's body.

from the above calculation, we can say that; the BaryCenter of the earth & moon is inside the earth that's why the earth is wobbling around it. & moon not rotating around the earth, it is rotating around the BaryCenter of the earth & moon.