DOPPLER SHIFT AND THE EARTH’S VELOCITY
PURPOSE: To show that the study of spectral lines can lead to the determination of velocities.
PROCEDURE: Determine the Doppler shift in the spectral lines of a star, observed at a six
month interval, and determine the velocity of the earth and the distance to the sun..
The doppler effect is named after Christian Doppler, who first noted the effects on sound waves. Later, Fizeau applied the Doppler effect to light waves and recognized its importance in astronomical applications.
According to the Doppler principle, an object which is moving towards us will produce spectral lines shifted towards the blue end of the spectrum. An object receding from us will show red shifted lines. The amount of the shift (！，) determines the velocity of the body according to
where v is the velocity of the object, c the velocity of light, ， the normal rest position of the line
observed, and ！， the displacement of the line from its normal position. Two examples are given below, one for an approaching cloud of interstellar gas, the other for a receding cloud of gas. The laboratory comparison spectrum (or rest spectrum) is between the spectra of the clouds.
The Doppler effect only gives the velocity of the object in the line of sight. If the object is not moving radially with respect to the earth , then only the component of velocity in the line of sight can be determined.
EARTH’S ORBITAL VELOCITY
Let us consider the above diagram where the earth is regarded as moving in a circular orbit about the sun. The star lies in the plane of the ecliptic and is assumed to be stationary at an indefinitely great distance from the sun. At point A in its orbit, the earth is moving directly away from the star, but half a year later, at point B, the earth is moving directly toward the star.
Because of the Doppler effect, when the earth is at A, the star’s spectral lines will be shifted to longer wavelengths (red shift) by an amount corresponding to the earth’s orbital velocity. At B, the shift will be to shorter wavelengths (blue shift) by the same amount. In our ideal case, the measured Doppler velocity would be equal to the earth’s orbital velocity. In an actual application of the method, there are some complications:
1. The star may not be stationary with respect to the sun, but have some radial velocity
2. The star may not lie in the plane of the ecliptic.
3. Our earth’s orbit is slightly elliptical, with the result that its velocity is about 3.4%
greater at perihelion than at aphelion.
4. The rotation of the earth also gives the observer a velocity toward or away from the
star, but this is small compared to the velocity of revolution.
In this exercise, we will ignore the last two complications but correct for the first two as follows.
In dealing with line of sight motions, astronomers call the velocities of approach negative (-) and velocities of recession positive (+). Let (V) be the orbital velocity of the earth in km/sec, O
and (V) be the radial velocity of the star relative to the sun. Then at time A, the motion of the S
VVV，？star relative to the earth is . Similarly, at time B the velocity is . VVV，？BSOASO
Solving for this pair of equations, we find the earth’s velocity is
and for the star
If the star is not in the plane of the ecliptic, but is at a celestial latitude (L), the value of V just O
obtained should be corrected for by dividing it by the cosine L.
The observational material for this exercise consists of two spectrograms of Arcturus taken about one half year apart on July 1, 1939 (a) and on January 19, 1940 (b). From the spectrum of (a) we can get Vand from (b) we can get V. A B
The bright lines seen above and below the star’s spectra are iron lines from a laboratory source impressed on the original plates at the time of observation to provide reference wavelengths. Corresponding to each of these comparison lines is a dark line in the stellar spectrum, displaced slightly to the right in (a) and to the left in (b).
The seven comparison line are labeled; their unshifted (rest) wavelengths in Angstroms are: 1. 4260.48 2. 4271.16 3. 4271.76 4. 4282.41 5. 4294.14 6. 4299.24 7. 4307.91.
The first thing you need to know is the scale of the spectrogram. Measure carefully, to a
0.1 mm, the distance between two widely spaced comparison lines. Divide that distance into the difference in angstroms between the two lines and this gives the scale of the spectrogram in Å/mm.
Next, select three lines and measure their displacements on (a) and (b). Take a few measurements of each so that an average value can be determined. The values will be small,
so be careful; a small error can lead to a large percentage error! Apply the proper algebraic sign
to each measured displacement, (+) for red shifted and (-) for blue shifted. RECORD ALL DATA IN TABLE FORM. You now have the displacements measured in mm but you need to know them in units of Angstroms (Å). Using the scale factor you determined earlier, the
displacement in Angstroms can be found. RECORD THE DATA IN THE TABLE.
Next, calculate V and V for the three lines you have measured. Record them in AB
your table. From the three values of V and V, find the average of each. Using these AB
average values, calculate V and V. The results of V and V should be similar, if there are OSAB
large differences between the values than there is an error in your work.
Because Arcturus is not in the plane of the ecliptic, V must be corrected for by dividing it O
by the cosine of the latitude of the star. For Arcturus, cosine L = 0.86.
If the earth’s orbital velocity is known, then the distance from the sun can be found from:
where P is the period in seconds and R is the average distance of the earth from the sun in kilometers.
The number of seconds in a year = 31,556,926.
1. What is the orbital velocity of the earth? How does your value compare to the accepted value?
What are the sources of error?
2. What is the radial velocity of the star? Is it approaching or receding from the sun? 3. What is the average distance to the sun? How does your value compare to the accepted value?
What are the sources of error?
4. Could the rotation of the sun be determined by using the Doppler effect? Explain. 5. Could the rotation of a star be determined by using the Doppler effect? Explain.