4) Say we want to measure whether a star is coming towards us or away from us using the Doppler shift of the lines in its spectra, with an accuracy of 1 km s-1 or better. Assume for simplicity that the star is in the plane of the Earth's orbit around the Sun. (a) Is it important to take into account the Earth's orbital motion around the Sun? How about Earth's rotational motion on its axis (ignore the tilt of the Earth's rotation axis for this problem)? In other words, if we didn't take these motions into account, could we make an error larger than 1 km s-1 ? Use figures to show these effects and calculate what are the largest velocity shifts these two effects can introduce, in km s-1 ? Use the Earth’s sidereal year, orbit size, sidereal day, and radius. (b) For the Earth's rotational motion, describe qualitatively how the velocity shift depends on latitude. That is, does it increase, decrease, or stay the same, and why?

Explanation:

we have that angular velocity that ω = 2πf

where f = 1/t, and t = 3.15 × 10⁷ sec/year

But the Earths revolution speed around the sun is given about 29.78 km/s

this velocity V₁ = 2πr/T

where T = 1 year

And the Earth rotates at a speed of 360° in 1 day(24 hrs)

from the angular velocity ω = 2πf

ω = 7.292 × 10⁻5 rad/sec  (1)

The maximum distance from axis of rotation of the earth at the equator is given req = 6378 m

Measuring the Shifts as a result of the Earth surface movement at equator;

V₂ = ωr

V₂ =  7.292 × 10⁻5 ˣ 6378 = 0.465 km/s

V₂ = 0.465 km/s

From this we have that the maximum effect of the Earth's motion around the sun and it's rotation will be

Vt = V₁ + V₂

Vt = 29.72 + 0.465 = 30.2 km/s

Vt = 30.2 km/s

(b). As the latitude increases, the velocity shift decreases.

Explanation:

Moving away from the equator towards the pole direction, we figure that the distance of the surface decreases with rotation speed. That is to say that at the poles we have a zero value and at the equator we get a maximum value.

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