= 17: Doppler shift of a spectral line: One of the most prominent spectral lines of...

Transcribed Image Text:

= 17: Doppler shift of a spectral line: One of the most prominent spectral lines of hydrogen is the H line, a bright red line with a wavelength of \o 656.1 10-9 m. Note: The speed c, wavelength \ and frequency of light are related by c = - . Q17a: What is the wavelength of the Ha line from a star receding with a speed of 3000 km/s? Hint: First calculate and then plug into the Doppler shift formula, keeping in mind that when the frequency goes down, the wavelength goes up. 817b: The Ha line measured on Earth from opposite ends of the Sun's equator differ in wavelength by A = 9 10-12 m. Assuming that the effect is caused by the rotation of the Sun, find the period of rotation (in days). The diameter of the Sun is d = 1.39 106 km. Hint: The light from one end is blue shifted and the light from the other end is red shifted. When is small, you can simplify the algebra by Taylor expanding the longi- tudinal Doppler shift formula and keeping just the terms upto linear order in . Find an expression for A in terms of and B. Since you are told A and > 0, you now know and therefore v. Then, write the period of rotation of the sun, T, in terms of its diameter and the surface speed, v, at the equator. Plug in the expression for v you got before, to get an expression for T in terms of Earth's diameter, A and AA. Q17c: Besides the longitudinal Doppler shift coming from the rotation of the sun, there is also a transverse Doppler shift due to the revolution of Earth around the sun. The speed of Earth is 30 km/s. Estimate the magnitude of the transverse Doppler shift to see if we were justified in ignoring it in Q17b. (You can answer this consider- ing either the shift in wavelength or the shift in frequency.) Hint: There are 2 factors to consider here. First, and more importantly, when is small, think about how the leading order term in the longitudinal and transverse Doppler shift formulae depends on . Second, the speed of rotation of the surface of the sun, versus the speed of the Earth in its orbit around the sun. = 17: Doppler shift of a spectral line: One of the most prominent spectral lines of hydrogen is the H line, a bright red line with a wavelength of \o 656.1 10-9 m. Note: The speed c, wavelength \ and frequency of light are related by c = - . Q17a: What is the wavelength of the Ha line from a star receding with a speed of 3000 km/s? Hint: First calculate and then plug into the Doppler shift formula, keeping in mind that when the frequency goes down, the wavelength goes up. 817b: The Ha line measured on Earth from opposite ends of the Sun's equator differ in wavelength by A = 9 10-12 m. Assuming that the effect is caused by the rotation of the Sun, find the period of rotation (in days). The diameter of the Sun is d = 1.39 106 km. Hint: The light from one end is blue shifted and the light from the other end is red shifted. When is small, you can simplify the algebra by Taylor expanding the longi- tudinal Doppler shift formula and keeping just the terms upto linear order in . Find an expression for A in terms of and B. Since you are told A and > 0, you now know and therefore v. Then, write the period of rotation of the sun, T, in terms of its diameter and the surface speed, v, at the equator. Plug in the expression for v you got before, to get an expression for T in terms of Earth's diameter, A and AA. Q17c: Besides the longitudinal Doppler shift coming from the rotation of the sun, there is also a transverse Doppler shift due to the revolution of Earth around the sun. The speed of Earth is 30 km/s. Estimate the magnitude of the transverse Doppler shift to see if we were justified in ignoring it in Q17b. (You can answer this consider- ing either the shift in wavelength or the shift in frequency.) Hint: There are 2 factors to consider here. First, and more importantly, when is small, think about how the leading order term in the longitudinal and transverse Doppler shift formulae depends on . Second, the speed of rotation of the surface of the sun, versus the speed of the Earth in its orbit around the sun.