Purpose:
This experiment will try to measure the orbital velocity of the Earth using the Doppler shift measurements in the spectrum of the star, Arcturus. Moreover, this experiment will become a measurement of the accuracy of such technique in measuring the orbital velocity of the Earth. Overall, the experiment will familiarize the students with the concept of Doppler shift and other related concepts.
Procedure:
This experiment was performed using personal measurements from the given two spectrograms of the Arcturus specifically 6 months apart. The upper spectrogram labeled “a” from the given spectrograms represents the red-shifted spectrogram while the bottom spectrogram labeled “b” represents the blue-shifted spectrogram. The shifts from the lines were measured using a ruler and a magnifying glass. The measured shifts were recorded and subjected to calculations.
As for the conversion factor from millimeters to angstroms, the length from reference line 1 to reference line 7 was measured and recorded. Since the actual distance of reference line 1 to reference line 7 in angstroms is known, the conversion factor from millimeters to angstroms was calculated and recorded.
Figure 1. The spectrograms of Arcturus: the red-shifted spectrogram (A) and the blue-shifted spectrogram (B)
Data and Calculation
First, the conversion factor from millimeters to angstroms was calculated. The measured distance from reference line 1 to reference line 7 was 143 millimeters. The actual wavelengths of lines 1 and 7 are 4260.48 angstroms and 4307.91 angstroms, respectively. Therefore, the conversion unit can be calculated by dividing the actual distance from line 1 to line 7 with the measured distance in millimeters.
Conversion Factor CF= Actual distance (angstroms)Measured distance (millimeters)
(Equation 1)
CF= 4307.91 angstroms-4260.48 angstroms143 millimeters=0.33168 angstroms/millimeter
The measured shifts of spectrograms A and B from the reference lines were denoted as Δλ. Since they are measured in millimeters, they had to be converted into angstroms using the conversion factor that was earlier derived. Equation 2 shows the conversion process.
Δλ angstroms = Δλ millimeters×CF
(Equation 2)
As the shift is converted into angstroms, the relative velocity between the Earth and Arcturus can be calculated using the equation below.
Vr=Δλλ0c
(Equation 3)
In Equation 3, λ0 refers to the wavelength of the corresponding reference line while c refers to the speed of light which is 300,000 kilometers per second. Table 1 shows the recorded shifts, the converted measurements into angstroms, and the relative velocity between the Earth and Arcturus according to the corresponding values.
The average relative velocity was computed per spectrogram by dividing the sum of the computed relative velocities by 4.
Vrr=23.355+17.470+17.379+23.0984=20.326 km/s
Vrb=(-31.062)+(-23.293)+(-32.441)+(-30.720)4=-29.379 km/s
Vrr refers to the average relative velocity of the red-shifted spectrum while Vrb refers to the average relative velocity of the blue-shifted spectrum. Since Vrr = VStar + VEarth and Vrb = VStar - VEarth, we can solve for the orbital velocity of the Earth, VEarth.
VEarth=Vrr-Vrb2=20.326--29.3792kms=24.852kms
Conclusion and Analysis
Since Arcturus is not located at ecliptic plane, the true orbital value can be computed by dividing the computed orbital velocity by the cosine of the latitude where Arcturus lies (31˚).
Orbital Velocity =24.852cos31°=28.898 kilometers per second
The theoretical orbital velocity of the Earth is 29.78 kilometers per second, which means that the computed value of 28.898 kilometers per second has an error of 2.96%. The error is most probably from measurement errors, especially during the measurements of the shifts from the reference lines. Still, the computed value is close to the theoretical value suggesting that the experiment is a success.
