(a) The maximum radial velocity of a star with an orbiting exoplanet is given by Equation (5.8) for an edge-on orbit. Us
Posted: Fri Apr 29, 2022 12:04 pm
(a) The maximum radial velocity of a star with an orbiting
exoplanet is given by Equation (5.8) for an edge-on orbit. Use this
equation to estimate the radial velocity signature an alien would
see if they tried to detect Jupiter orbiting the Sun at 5 au.
(b) Estimate the maximum radial velocity of a Sun-like star with
an Earth-mass companion at 1 au, in an edge-on orbit.
(c) The first radial velocity measurements (Mayor & Queloz
1995) could detect stellar velocities of about 10 m s−1.
Would they have been sensitive enough to detect a Jupiter-like
planet? An Earth-like planet?
(d) A modern spectrograph, like the High Accuracy Radial
Velocity Planet Searcher (HARPS; Mayor et al. 2003) can detect
stellar velocities of about 1 m s−1. Is it sensitive
enough to detect a Jupiter-like planet? An Earth-like planet?
(e) Figure 5.5 shows a portion of the solar spectrum that
includes the sodium D lines–one of which is centered at 5889.950
Angstroms. Given the radial velocity you calculated above for a
Jupiter-like planet, and the Doppler shift equation (Equation
(5.1)), what would be the expected wavelength shift for this line?
What would be the expected wavelength shift for an Earth-like
planet? Evaluate the size of your answer by comparing to the sizes
of physical objects.
(f) The resolution of a spectrograph is measured by its spectral
resolution, R=λΔλ, where Δλ is the smallest wavelength shift that
can be measured, and λ is the wavelength observed. For the HARPS
instrument, which has a spectral resolution of R = 120,000, what is
Δλ when observing near the 5889.950 Angstrom Sodium D line?
(g) Discuss how the minimum wavelength shift detectable by HARPS
compares with the expected wavelength shift for a Jupiter-like
planet, and for an Earth-like planet. Based on what you find,
discuss why exoplanet researchers may need to measure the shifts of
a large number of lines, rather than just one.
тр 2л- M* -ap 7. = 2ла р 3/2 (5.7) (GM, G = тр, (5.8) Мар
Ό, Δλ = λο C (5.1)
0.30 T 0.25 0.20 Intensity [W cm-1 ster-1 Å-1] m 0.15 0.10 0.05 0.00 5800 5825 5975 6000 5850 5875 5900 5925 5950 Wavelength [Angstroms] Figure 5.5. A collapsed, one-dimensional spectrum of the Sun from 5800 to 6000 Å, showing the many absorption bands arising from atoms in the Sun's atmosphere (Brault & Neckel 1987). The two deep lines at 5889.950 and 5895.924 Å are due to absorption from sodium, and are known as the “Sodium D lines”. If you've ever placed sodium in a Bunsen burner in Chemistry class and seen the bright yellow flame that results, you've seen emission in these same bands.
exoplanet is given by Equation (5.8) for an edge-on orbit. Use this
equation to estimate the radial velocity signature an alien would
see if they tried to detect Jupiter orbiting the Sun at 5 au.
- A The Maximum Radial Velocity Of A Star With An Orbiting Exoplanet Is Given By Equation 5 8 For An Edge On Orbit Us 1 (22.58 KiB) Viewed 34 times
an Earth-mass companion at 1 au, in an edge-on orbit.
(c) The first radial velocity measurements (Mayor & Queloz
1995) could detect stellar velocities of about 10 m s−1.
Would they have been sensitive enough to detect a Jupiter-like
planet? An Earth-like planet?
(d) A modern spectrograph, like the High Accuracy Radial
Velocity Planet Searcher (HARPS; Mayor et al. 2003) can detect
stellar velocities of about 1 m s−1. Is it sensitive
enough to detect a Jupiter-like planet? An Earth-like planet?
(e) Figure 5.5 shows a portion of the solar spectrum that
includes the sodium D lines–one of which is centered at 5889.950
Angstroms. Given the radial velocity you calculated above for a
Jupiter-like planet, and the Doppler shift equation (Equation
(5.1)), what would be the expected wavelength shift for this line?
What would be the expected wavelength shift for an Earth-like
planet? Evaluate the size of your answer by comparing to the sizes
of physical objects.
- A The Maximum Radial Velocity Of A Star With An Orbiting Exoplanet Is Given By Equation 5 8 For An Edge On Orbit Us 2 (6.97 KiB) Viewed 34 times
- A The Maximum Radial Velocity Of A Star With An Orbiting Exoplanet Is Given By Equation 5 8 For An Edge On Orbit Us 3 (168.76 KiB) Viewed 34 times
resolution, R=λΔλ, where Δλ is the smallest wavelength shift that
can be measured, and λ is the wavelength observed. For the HARPS
instrument, which has a spectral resolution of R = 120,000, what is
Δλ when observing near the 5889.950 Angstrom Sodium D line?
(g) Discuss how the minimum wavelength shift detectable by HARPS
compares with the expected wavelength shift for a Jupiter-like
planet, and for an Earth-like planet. Based on what you find,
discuss why exoplanet researchers may need to measure the shifts of
a large number of lines, rather than just one.
тр 2л- M* -ap 7. = 2ла р 3/2 (5.7) (GM, G = тр, (5.8) Мар
Ό, Δλ = λο C (5.1)
0.30 T 0.25 0.20 Intensity [W cm-1 ster-1 Å-1] m 0.15 0.10 0.05 0.00 5800 5825 5975 6000 5850 5875 5900 5925 5950 Wavelength [Angstroms] Figure 5.5. A collapsed, one-dimensional spectrum of the Sun from 5800 to 6000 Å, showing the many absorption bands arising from atoms in the Sun's atmosphere (Brault & Neckel 1987). The two deep lines at 5889.950 and 5895.924 Å are due to absorption from sodium, and are known as the “Sodium D lines”. If you've ever placed sodium in a Bunsen burner in Chemistry class and seen the bright yellow flame that results, you've seen emission in these same bands.