The purpose of this experiment is to find the orbital velocity of the Earth by measuring the Doppler shift in the spectrum of the star Arcturus.
Diagram showing the earth and Arcturus
An Explanation of the Procedure
Line 1 was selected and its displacement was measured in (a) and (b) in millimeters. A straight edge was used to connect the matching reference lines as given in the above figure. The shifts were measured with a scale to a tength of millimeter. This measurement gave the value to the shift Δλ for the line in millimeter. A table was created, as shown below, where all the readings were recorded. The process was repeated for the lines 1, 3, 5, and then 7. All measurements were recorded in the table. The conversion factor used in obtaining the wavelength shift in angstroms was calculated by dividing the difference in the actual wavelength of line 1 and line 7 by the distance between 1 and 7.
Data and Calculations
The Conversion Factor was calculated using (4307.91-4260.48)/85= 0.558A/mm
This value gotten makes the average shift in spectrum (a) to be 0.1913A and that of spectrum (b) is 0.4384A.
The relative velocity can then be calculated using the formula, V=c Δλ/ λ0 and then solving the system of equations: Va = (Ve-V0) and Vb=-(Ve+V0). The average velocity of spectra (a) has been found to be 13.3968 Km/s while the average velocity of spectra (b) has been found to be 30.7012 Km/s.
The vrr= 19.46855 km/s
vrr= vstar+ vearth
vrb= -29.1441 km/s
vrb= vstar - vearth
Since vrr= vstar+ vearth
Therefore, vstar= vrr - vearth
Putting this into vrb= vstar - vearth
vrb= vrr - vearth - vearth
Therefore, vearth= (vrr - vrb)/2
When this system is solved, the velocity of Acturus relative to the sun is about 22.05 Km/s i.e. (13.3968 + 30.7012)/2.
Since the Arcturus is assumed to be located at the ecliptic plane, then, true orbital is 24.306/0.86 = 28.263 km/s. With this, the length of 1AU is calculated to be 3.0x107Km.
The percentage difference is calculated using the formula (calculated – accepted)/(calculated + accepted) * 200 = (28.263 – 30)/( 28.263 + 30)= 5.96%
Conclusion and Analysis
A small red shift was observed in spectra (a) while a large blue shift was observed in spectra (b). From these two observations it can be inferred that the movement of the Arcturus is towards the earth.
This experiment has an acceptable level of experimental uncertainty. The initial calculations gave an error of 0.4mm which falls between the acceptable ranges and the final uncertainty stayed at 5.96%.
The value of the 1AU gotten in this experiment falls within the permitted values of the experimental uncertainty.
The two possible source of error in this experiment include inaccurate time measurement and error due to parallax. A more accurate result could be achieved if the initial measurements were done using finer techniques.
