How astronomer could detect a large planet orbiting a star without actually seeing the planet?

Can you think of any way an astronomer could detect a large planet orbiting a star without actually seeing the planet? (Hint: How would the star move as the planet orbits it?)How astronomer could detect a large planet orbiting a star without actually seeing the planet?The planet could cause the star to "wobble" from our point of view as the planet's gravity pulled first one and then the other as the planet orbited the star. This method is called radial velocity or Doppler method.



The planet will also cause the star's position in the sky to change over time (it will actually move in a small ellipse). This is the astrometry method.



The planet could also cross in front of the star, causing its light to dim a bit. This is the transit method.How astronomer could detect a large planet orbiting a star without actually seeing the planet?
I was using my answer as a kind of scratch paper, trying to work out the math. I left it midway through and never got back to finish. The V near the end of my answer is not the same as the V defined at the beginning, and I should have used some other symbol, like a lower-case v.

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How astronomer could detect a large planet orbiting a star without actually seeing the planet?A large planet will have enough gravitational influence on the star that it will cause very slight movement of the star in a small circle. They both orbit the barycenter, or the gravitational center of the system, which would be slightly off center of the star. We can detect small motions of the star. From this we can also measure the planet's orbital period. By knowing the kind of star and getting an estimate of its mass, we can then figure out the approximate mass of the planet.How astronomer could detect a large planet orbiting a star without actually seeing the planet?
GravityHow astronomer could detect a large planet orbiting a star without actually seeing the planet?M = mass of star

m = mass of planet

R = distance of star from barycenter

r = distance of planet from barycenter

P = period of orbit (observed)

V' = half the observed difference between the star's radial speed at maximum and minimum (observed)

Q = angle between the line of sight and the planet's angular momentum vector



V = V' / sin Q = orbital speed of star, corrected for aspect angle



Assuming you can determine Q somehow, then, assuming the relative orbit is a circle (big assumptions here)...



V = 2 pi R / P

R = V P / (2 pi)



P^2 = 4 pi^2 R^3 / [G (M+m)]

M+m = 4 pi^2 R^3 / (G P^2)

M+m = V^3 P / (2 pi G)



The calculated sum of the masses depends on the cube of the sine of the aspect angle. So it's important to know Q accurately. I think you'd probably have to see the shape of the orbit over time to get a reasonably good value for Q, so the transverse motion also must be measured.



We may combine the distance modulus with mass luminosity relation for main sequence stars having masses between 0.8 and 1.4 suns, we get



M = (5.94E+30 kg) sqrt(d/10pc) 10^{-q/10}



Where d is the distance to the star, perhaps obtained by parallax, and q is the star's apparent magnitude.



V = sqrt{ G (M + m) / (R + r) }

R + r = G (M+m) / V^2



r = (M/m) R



R (1 + M/m) = G (M+m) / V^2