Universe
Universe
11th Edition
ISBN: 9781319039448
Author: Robert Geller, Roger Freedman, William J. Kaufmann
Publisher: W. H. Freeman
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Chapter 17, Problem 80Q
To determine

Whether the orbital speed of earth or the speed of the earth about its axis or both is accountable for the study of radial velocities of the stars or not. Also, determine the value of the orbital speed and the speed of the earth about its own axis. Also, explain how astronomers compensated the effect of earth orbital speed and its speed about its axis.

Expert Solution & Answer
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Answer to Problem 80Q

Solution:

vorbital=29.8 km/s, vrotational=463 m/s, and astronomers feed these data into the modern telescope to compensate the effect of orbital and rotational speed of the earth.

Explanation of Solution

Given data:

The stars radial velocity is to be measured to an accuracy of 1 km/s or better.

Formula used:

The formula for radius of a sphere in term of diameter is:

r=D2

Here, D is the diameter.

The expression used for the circumference of a circle C of radius r is,

C=2πr

The expression used for the speed v of an object is,

v=distancetime

Explanation:

Refer to the appendix 1, the value of the orbital radius of the earth is,

r=149.6×106 km

The expression for the circumference at the orbit of the earth is,

C=2πr

Substitute 149.6×106 km for r.

C=2π(149.6×106 km)=9.4×108 km

Use the expression of speed to write the expression of orbital speed of the earth.

vorbital=Ct

Substitute 9.4×108 km for C and 1 yr for t.

v=9.4×108 km1 yr=9.4×108 km1 yr(3.154×107 s1 yr)=29.8 km/s

Refer to the appendix 2 to obtain the value of the diameter of the earth:

D=12756 km

The formula for radius of earth is:

r=D2

Substitute 12756 km for D.

r=12756 km2=6378 km

Therefore, the radius of the earth is 6378 km.

The expression for the circumference of the earth at equator is:

C=2πr

Substitute 6378 km for r.

C=2π(6378 km)=4.0×104 km

Use the expression of speed to write the expression of rotational speed of the earth.

vrotational=Ct

Substitute 4.0×104 km for C and 1 day for t.

vrotational=4.0×104 km1 day=4.0×104 km(1000 m1 km)1 day(86400 s1 day)=463 m/s

The orbital speed of the earth is 29.8km/s. So, it will impact the calculation of the radial velocity of the star if the observation is done from the surface of the earth. Also, the rotational speed of the earth is 463 m/s, which is also large enough to impact the observation for the calculation of radial velocity of the earth. So, to obtain the value of the radial velocity with accuracy upto 1 km or better, the details of the orbital and rotational speed of the earth is feed to the telescope.

Conclusion:

The orbital speed of the earth is 29.8km/s and its rotational speed is 463 m/s, which is sufficient to impact the calculation for the radials velocity of the stars. So, these are taken into consideration for the calculation purpose.

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Universe

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