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Consider an electron gyrating in the magnetic field associated with a sunspot which has a magnetic field strength of 0.25 T. a) Compute the cyclotron frequency we of the electron in Hertz. b) The typical temperature of a sunspot is 3500 K. Use this temperature and the equipartition equation for velocity v, temperature T and mass m of a particle, that is, 3KBT V = m to compute the velocity of the electron in meters per second. Here, kB is Boltzmann's constant (1.38 x10-23 J/K). c) Calculate the Larmor radius TL of gyration of the electron. Compute and physically interpret the ratio of this radius to the radius of the Sun (6.96×10 m). d) Calculate the centripetal acceleration a in m/s² of the electron as it gyrates along the magnetic field lines associated with the sunspot. e) Calculate the power P in Watts emitted by this electron as it gyrates along the magnetic field lines associated with the sunspot. f) Repeat Parts (a) through (e) for a proton. g) Compute the ratio of the power emitted by the electron to the power emitted by the proton. Therefore, is the cyclotron emission detected from the Sun dominated by emission from electrons or protons? Explain vour answer.

Consider an electron gyrating in the magnetic field associated with a sunspot which has a magnetic field strength of 0.25 T. a) Compute the cyclotron frequency we of the electron in Hertz. b) The typical temperature of a sunspot is 3500 K. Use this temperature and the equipartition equation for velocity v, temperature T and mass m of a particle, that is, 3KBT V = m to compute the velocity of the electron in meters per second. Here, kB is Boltzmann's constant (1.38 x10-23 J/K). c) Calculate the Larmor radius TL of gyration of the electron. Compute and physically interpret the ratio of this radius to the radius of the Sun (6.96×10 m). d) Calculate the centripetal acceleration a in m/s² of the electron as it gyrates along the magnetic field lines associated with the sunspot. e) Calculate the power P in Watts emitted by this electron as it gyrates along the magnetic field lines associated with the sunspot. f) Repeat Parts (a) through (e) for a proton. g) Compute the ratio of the power emitted by the electron to the power emitted by the proton. Therefore, is the cyclotron emission detected from the Sun dominated by emission from electrons or protons? Explain vour answer.

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Only answer the second question, the first is referenced in question #2

**Problem Set: Sun's Properties and Sunspots** This exercise involves calculations related to the Sun, its gravitational effect, and the properties of sunspots. Follow the steps outlined below to solve the problems: a) **Compute Acceleration Due to Gravity at the Sun's Surface:** - Given: The mass of the Sun \( M = 2 \times 10^{30} \) kg. - Use this information and details from Problem 1 to calculate the gravitational acceleration \( g \) at the Sun's surface. b) **Calculate Drift Velocity Due to Gravity:** - Calculate the drift velocity \( \vec{v}_G \) for a proton and an electron located within a sunspot, influenced by the Sun's gravity. c) **Determine Density of Protons and Electrons in the Photosphere:** - Assume a typical mass density of the Sun's photosphere of \( 10^{-3} \text{ kg/m}^3 \). - Calculate the number density of protons and electrons, considering the Sun is composed of pure ionized hydrogen. d) **Compute Total Current Density in Sunspots:** - Calculate the total current density \( \vec{J} \), taking into account the contributions of both electrons and protons within sunspots. - Assume a fully-ionized hydrogen gas where the electron density \( n_e \) and proton density \( n_p \) are equal. These calculations will help you understand various properties related to the dynamics within the Sun's atmosphere, particularly how particles behave within sunspots.
Transcribed Image Text:**Problem Set: Sun's Properties and Sunspots** This exercise involves calculations related to the Sun, its gravitational effect, and the properties of sunspots. Follow the steps outlined below to solve the problems: a) **Compute Acceleration Due to Gravity at the Sun's Surface:** - Given: The mass of the Sun \( M = 2 \times 10^{30} \) kg. - Use this information and details from Problem 1 to calculate the gravitational acceleration \( g \) at the Sun's surface. b) **Calculate Drift Velocity Due to Gravity:** - Calculate the drift velocity \( \vec{v}_G \) for a proton and an electron located within a sunspot, influenced by the Sun's gravity. c) **Determine Density of Protons and Electrons in the Photosphere:** - Assume a typical mass density of the Sun's photosphere of \( 10^{-3} \text{ kg/m}^3 \). - Calculate the number density of protons and electrons, considering the Sun is composed of pure ionized hydrogen. d) **Compute Total Current Density in Sunspots:** - Calculate the total current density \( \vec{J} \), taking into account the contributions of both electrons and protons within sunspots. - Assume a fully-ionized hydrogen gas where the electron density \( n_e \) and proton density \( n_p \) are equal. These calculations will help you understand various properties related to the dynamics within the Sun's atmosphere, particularly how particles behave within sunspots.
Consider an electron gyrating in the magnetic field associated with a sunspot which has a magnetic field strength of 0.25 T. a) Compute the cyclotron frequency \(\omega_c\) of the electron in Hertz. b) The typical temperature of a sunspot is 3500 K. Use this temperature and the equipartition equation for velocity \(v\), temperature \(T\) and mass \(m\) of a particle, that is, \[ v = \sqrt{\frac{3k_BT}{m}}, \] to compute the velocity of the electron in meters per second. Here, \(k_B\) is Boltzmann’s constant (1.38 \(\times 10^{-23}\) J/K). c) Calculate the Larmor radius \(r_L\) of gyration of the electron. Compute and physically interpret the ratio of this radius to the radius of the Sun (6.96 \(\times 10^8\) m). d) Calculate the centripetal acceleration \(a\) in m/s\(^2\) of the electron as it gyrates along the magnetic field lines associated with the sunspot. e) Calculate the power \(P\) in Watts emitted by this electron as it gyrates along the magnetic field lines associated with the sunspot. f) Repeat Parts (a) through (e) for a proton. g) Compute the ratio of the power emitted by the electron to the power emitted by the proton. Therefore, is the cyclotron emission detected from the Sun dominated by emission from electrons or protons? Explain your answer.
Transcribed Image Text:Consider an electron gyrating in the magnetic field associated with a sunspot which has a magnetic field strength of 0.25 T. a) Compute the cyclotron frequency \(\omega_c\) of the electron in Hertz. b) The typical temperature of a sunspot is 3500 K. Use this temperature and the equipartition equation for velocity \(v\), temperature \(T\) and mass \(m\) of a particle, that is, \[ v = \sqrt{\frac{3k_BT}{m}}, \] to compute the velocity of the electron in meters per second. Here, \(k_B\) is Boltzmann’s constant (1.38 \(\times 10^{-23}\) J/K). c) Calculate the Larmor radius \(r_L\) of gyration of the electron. Compute and physically interpret the ratio of this radius to the radius of the Sun (6.96 \(\times 10^8\) m). d) Calculate the centripetal acceleration \(a\) in m/s\(^2\) of the electron as it gyrates along the magnetic field lines associated with the sunspot. e) Calculate the power \(P\) in Watts emitted by this electron as it gyrates along the magnetic field lines associated with the sunspot. f) Repeat Parts (a) through (e) for a proton. g) Compute the ratio of the power emitted by the electron to the power emitted by the proton. Therefore, is the cyclotron emission detected from the Sun dominated by emission from electrons or protons? Explain your answer.
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