What does the orbit look like if the planet has more mass and can revolve around sun?

[pruthvi]

consider a giant planet have a 3/4 mass of it sun and with the distance of 50AU from the sun.
does the planet revolve around in a elliptical orbit because the mass of the planet is high so there barycenter is half the distance between them so it look like binary star system but the question is as per Kepler rules the plant should revolve around the sun in a elliptical orbit. How it is possible in this case to say that the planet is in elliptical orbit ?

 

[Orodruin]

Kepler’s laws were empirically determined for planets with much smaller mass than the primary. They are not directly applicable to the case where the masses are similar.

However, the two body problem may be reformulated as a Kepler problem for the reduced mass in a central gravitational potential given by the same expression as for the two-body problem. This means that Kepler’s laws do hold true, albeit with a different mass relation. In particular, the orbit is indeed elliptical, but relative to the barycenter.

 

[Baluncore]

consider a giant planet have a 3/4 mass of it sun and with the distance of 50AU from the sun.

That would make a binary star. It would rotate about a barycenter somewhere near the middle.

How it is possible in this case to say that the planet is in elliptical orbit ?

It would not be considered to be a planet.

 

[Janus]

Every planet orbits around it's barycenter with the Sun. In most cases, this barycenter is close enough the center of the Sun to make little difference. The Jupiter-Sun barycenter is actually just above the surface of the Sun. There is no real difference between Earth orbiting the Sun and Jupiter other than the size of the Sun's orbit around the respective barycenters. Kepler's Laws reflected a pattern he derived via observation, and of course, were limited by those observations. The small variation due to planets orbiting barycenters rather than the Sun proper were more than likely within the error bars of his observations. Remember, he produced these laws before Newton came up with the theory of gravity which explained them.

 

[snorkack]

We cannot see the barycenter of a binary star. It is an unmarked dark spot in empty sky between the stars. If you want to find out the actual location of the barycenter of the binary star, you must make great precision measurement of large angles from the binary star to fixed points in the sky... over a long, long time!
If you simply measure the separation between the stars and their position angle then both stars follow elliptical orbits - of equal size, because you have only one separation. And the stars are NOT required to be at the focus (because the orbit can have any inclination!).

Measuring these large angles is useful, though, if you can. You need to track the proper motion, too. Because if you find a position along the separation which follows a straight proper motion while the stars have wavy proper motion then this is the barycenter. And what it gives to you is the ratio of lever arms - the ratio of the masses of the components.

 

[Orodruin]

Every planet orbits around it's barycenter with the Sun.

In most cases the difference is negligible to the extent that other effects, such as the jovian influence on the orbit, have a larger impact. In the case of Jupite it is relatively accurate.

For the interested, there is a telling image on Wikipedia:

This displays the motion of the solar system barycenter relative to the Sun.

 

[Nik_2213]

Easy way to visualise it is for Earth/Moon system, whose barycentre, around which both components have elliptical orbits, is located on average 4,671 km (2,902 mi) from Earth's centre, which is 75% of Earth's radius of 6,378 km (3,963 mi).
https://en.wikipedia.org/wiki/Barycenter_(astronomy)

Note that solar 'tides', plus the gravitational effects of Jupiter, Venus etc etc do nudge / perturb the orbits, though the 'dominant' factor is 'receding' Moon due momentum exchange via tidal dissipation...