https://photos.app.goo.gl/FGb6LqhESERuZ6zg7
Assume that the earth speed v0 = 0. Star began to approach the earth at the speed of v. Star came to position B, and the elapsed time was T seconds. The gravitational changes experienced by the earth are as follows:

T = 0s: F = G * M * m / R^2

T = 1s: F = G * M * m / (R + 1000)^2 ≈ G * M * m / R^2

According to the law of conservation of momentum:

F * T = mv1-mv0 = mv1, then v1 = F * T / m = T * G * M / R^2

T = 0s: v1 = 2 * G * M / R^2

T = 1s: v1 = 1 * G * M / R^2

Because of the speed v of the star, the earth obtained a different speed v1. The larger v is, the larger v1 will be, but the gravitational force on the planet from the star has hardly changed. If the star's v = X * cos(wt), then the speed change of earth will also show volatility.

The relative speed between them really affects the speed of the earth. Do you still think there is no Doppler effect between them? If you understand the Doppler effect, you will know that it exists anywhere.