Is 0.026 the Key to Unveiling a New Exoplanet's Characteristics?

[shirin]

There is a problem:
"The distance of Barnard’s star is 1.83 pc and mass 0.135 M. It has been suggested that it oscillates with an amplitude of 0.026"

in 25 year periods. Assuming this oscillation is caused by a planet, find the mass and radius of the orbit of this planet."
My question is what is 0.026"? Is it the angular size of 2*a of the star's orbit?

 

[Astronuc]

There is a problem:
"The distance of Barnard’s star is 1.83 pc and mass 0.135 M. It has been suggested that it oscillates with an amplitude of 0.026"

in 25 year periods. Assuming this oscillation is caused by a planet, find the mass and radius of the orbit of this planet."
My question is what is 0.026"? Is it the angular size of 2*a of the star's orbit?

I imagine that the 0.026'' is 0.026 arcseconds as measured by the observer on earth. One has a distance to the star of 1.83 pc.

There are some symbols that appear as boxes.

 

[shirin]

I don't know what it means that "the star oscillates with an amplitude of 0.026 arcsec"?
Does it mean that the angular size of its elliptical orbit (2a) around center of mass is 0.026? or whatelse?

 

[Astronuc]

I don't know what it means that "the star oscillates with an amplitude of 0.026 arcsec"?
Does it mean that the angular size of its elliptical orbit (2a) around center of mass is 0.026? or whatelse?

If a star is moving back and forth against the field of stars, then it means the star is revolving about some moment around the center of mass. At one time it is in one position, then in half the period, it is 0.026 arcsec from that position - diametrically opposed. The shorter arm is the one between COM and heavier mass.

Knowing the arclength and distance, one can calculate the moment arm of the star, or distance from the system's COM.

 

[pmacias]

I would first clarify the units of 0.026". If it is referring to angular size, then it would be measured in arcseconds or degrees, and the information provided is incomplete. However, if it is referring to the amplitude of the oscillation, then it would be measured in astronomical units (AU) or meters, and the information provided is sufficient.

Assuming the latter, we can use the formula for orbital motion to calculate the mass and radius of the planet. The mass of the planet can be found by using the equation: M = (a^3 * m_star) / (P^2 * G), where a is the semi-major axis of the orbit, m_star is the mass of the star, P is the period of the orbit, and G is the gravitational constant. Plugging in the values given, we can solve for the mass of the planet.

Next, we can use Kepler's third law to calculate the radius of the planet's orbit. This law states that the square of the orbital period is proportional to the cube of the semi-major axis. Therefore, we can rearrange the equation to find the semi-major axis (a) and then use it to find the radius of the orbit.

It is important to note that these calculations assume a circular orbit, so if the orbit is elliptical, the results may vary. Further observations and measurements would be needed to confirm the mass and radius of the planet's orbit.