Astronomy 217

 
 

Prof. Andrew W. Steiner

 
 
Oct. 11, 2021
 
 

TA James Ternullo

Last Time

  • Blackbody radiation

Today

  • Solar Observations

The Visible Sun

  • We study the Sun for 2 reasons.
  • The Sun influences the Earth, providing it with sunlight, striking it with the solar wind.
  • The Sun is the nearest star, our best chance to study a star up close. It was the only star whose image can be resolved and from which we collect neutrinos.

Luminosity

  • Luminosity is the total energy radiated by the Sun.
  • The amount of Sun's energy reaching Earth is called the Solar constant.
  • The mean solar constant is about \( 1400~\mathrm{W}/\mathrm{m}^2 \).
  • The Sun’s Luminosity, \( L_{\odot} \), can be calculated from the Solar constant and the Earth’s orbital separation.
  • \( L_{\odot} =4 \times 10^{26}~\mathrm{W} \), equivalent to 100 billion megatons of TNT per second

Solar Spectrum

  • Sunlight also tells us the composition of the Sun.
  • Spectral analysis, beginning with Fraunhofer in 1814, exhibits lines from 67 different elements.
  • Among these is helium, which was discovered by Norman Lockyear in 1868 in the spectrum of the chromosphere.
  • Helium was isolated terrestrially in 1895.
  • More than 90% of the atoms are hydrogen, 9% He and 0.13% “metals”

Photosphere

  • The visible surface of the Sun, or any radiating body, is the photosphere. The photosphere occurs at the point in the star where the optical depth, \( \tau = \sigma(x) n(x) x \) approaches 1.
  • This filtered image of the Sun shows a sharp edge to the Sun, telling us the Sun’s photosphere is very thin.
  • The radius of the photosphere is 696,000 km (commonly equated to the radius of the Sun, \( R_{\odot} \)), but the photosphere is ~ 400-500 km thick.
  • A blackbody of T = 5700K best fits the Sun’s photosphere.

H- Ions

  • The thinness of the solar photosphere, and its relative independence of wavelength, are the result of H− ions that occur for 2500 K < T < 10000 K.
  • In this range of temperature, collisional ionization occurs for many metals with low ionization potentials, but not for H, with a 13.6 eV ionization potential.
  • Hydrogen atoms combine with free electrons to make H− ions, which have an ionization potential of 0.75 eV. Thus H− can absorb any photon whose energy is greater than 0.75 eV, corresponding to λ = 1.7 μm.
  • For lower densities, H + e− collisions do not occur frequently enough to compete with photo-ionization, making a sharp cutoff in radius to the H− opacity.

Limb Darkening

  • The reduced intensity around the edge, or limb, of the solar disk is called limb darkening.
  • As one moves away from the center of the solar disk, the radius reached for a given column depth is reduced by the oblique angle.
  • Since the temperature increases toward the center of the Sun, \( T_B<T_A \), are seen on the limb.
  • With \( I \propto T^4 \), the limb can be as much as 1/10 as bright.

Solar Granulation

  • A closer look at the photosphere reveals a pattern of granulations, typically 1000 km across, that change on a 10 minute timescale.
  • These granulations are the visible top layer of the convection zone, with areas of hotter upwelling material (\( 1~\mathrm{km}~\mathrm{s}^{-1} \)) surrounded by areas of cooler sinking material.

Solar Structure

  • The peaking of the convective zone through the photosphere hints at the complicated interior structure of the Sun, which we will cover later.
  • Today, we’re focused on the visible parts of the sun, the solar atmosphere.
  • Above the photosphere lie 3 progressively less dense regions, the Chromosphere, the Transition Zone and the Corona.

Solar Magnetism

  • Key to understanding the behavior of the Sun’s atmosphere is the motion of fluid in a magnetic field.
  • A moving charge in a magnetic field experiences the Lorentz Force $$ \vec{F}_L = q \vec{v} \times \vec{B} = q \left( \vec{v}_{\perp} + \vec{v}_{\parallel} \right) \times \vec{B} $$
  • Motion along the magnetic field is unchanged, but motion perpendicular causes an acceleration $$ a = \frac{q}{m} v_{\perp} B $$
  • This acceleration pushes the particle in helical orbit of radius $$ r_c = \frac{m v_{\perp}}{q B} $$ (the Larmor radius)

Magnetic Energy

  • A magnetic field with strength \( B \) has an energy density $$ E_B = \frac{B^2}{2 \mu_0} $$ and an equivalent pressure $$ P_B = \frac{B^2}{2 \mu_0} $$
  • In terms of a 1 Tesla field: $$ P_B = 4 \times 10^5~\mathrm{N}~\mathrm{m}^{-2} \left( \frac{B}{1~\mathrm{T}} \right) $$
  • For comparison, the thermal gas pressure is $$ P_{\mathrm{gas}} = n k T = 5 \times 10^{3}~\mathrm{N}~\mathrm{m}^{-2} \left( \frac{\rho}{10^{-4}~\mathrm{kg}~\mathrm{m}^{-3}} \right) \left( \frac{T}{6000~\mathrm{K}} \right) $$ The importance of magnetic phenomenon in the gas can be deduced by comparing \( P_B \) to \( P_{\mathrm{gas}} \)

Sunspots

  • One very visible consequence of solar magnetic fields are sunspots, which move across the face of the Sun.

Sunspots in Motion

  • The Solar and Heliospheric Observatory (SOHO) orbits at Earth’s L1 point, outside the Earth’s magnetosphere measuring the Sun’s magnetic field, corona, vibrations, and ultraviolet emissions.

Magnetic Kinks

  • Sunspots, which typically last from days to months, result from loops of the Sun’s magnetic field breaking through the photosphere.
  • Sunspots are linked, a leading spot has one or more trailing spots that follow it.

Magnetic Cooling

  • The cooler temperature in a sunspot results from the magnetic pressure.
  • If we set the gas pressure outside of the sunspot equal to the sum on magnetic and gas pressure inside, $$ n k T_s + B^2/\mu_0 = n K T_p $$ this gives a magnetic field strength of $$ B = \left[ 2 \mu_0 n k \left( T_p - T_s \right) \right]^{1/2} $$
  • For the photosphere, \( \rho=3.5 \times 10^{-4} ~\mathrm{kg}~\mathrm{m}^{-3} \), thus $$ n = \rho/(\mu m_p) = 2.1 \times 10^{23}~\mathrm{m}^{-3} $$ while \( T_p - T_s \approx 1800~\mathrm{K} \)
  • This tells us the magnetic field in a sunspot is ≈ 0.1 T, 10-100 times larger than the average solar surface field.

Differential Rotation

  • As a fluid body, the Sun is not constrained to rotate rigidly. It rotates differentially, with a period near the poles of 35 days, but 25.4 days at the equator.
  • The arrangement of sunspots originates from magnetic field lines distorted by this differential rotation.

Sunspot Cycles

  • The frequency of sunspots follows an 11 year cycle.
  • However, the maximum amplitude between cycles can vary significantly. During the Maunder minimum in the late 1600s, few, if any, sunspots occurred.

Sunspot Latitude

  • The location of sunspots also follows the 11 year cycle, with sunspots first appearing at high latitudes early in the cycle and gradually lower as the cycle progresses.
  • The cycle actually has a 22-year period, as the magnetic polarities reverse between northern and southern hemispheres every 11 years.

Chromosphere

  • Above the photosphere lies the chromosphere, with a density \( 10^4 \) lower that the photosphere.
  • The chromosphere is difficult to see directly against the brightness of the photosphere.
  • However sometimes, Moon covers photosphere and not chromosphere during eclipse.
  • One can also observe the chromosphere using filters to select narrow bands with strong photospheric absorption, leaving the chromospheric emission lines visible, like \( H_β \) or He, which was discovered in the chromosphere.

Chromospheric Features

  • Plages: regions of strong magnetic field near sunspots.
  • Prominences: cool clouds of gas following a magnetic field feature.
  • Filaments: long dark features, prominences seen from above.

Spicules

  • Small scale storms in the chromosphere emit spicules, short-lived, narrow jets of \( 10~\mathrm{km}~\mathrm{s}^{-1} \) gas, typically lasting a few minutes.
  • They appear as dark spikes sprouting up from the solar chromosphere, visible against the face of the Sun because they are cooler than the underlying photosphere.

Corona

  • The upper most layer of the Sun’s atmosphere is the solar corona.
  • Solar corona can be seen during eclipse if both photosphere and chromosphere are blocked or using a coronagraph.
  • The inner K corona (R < \( 2.5~R_{\odot} \)) exhibits a continuum spectrum with emission lines, but no detectable absorption lines.
  • The outer F corona exhibits a continuum with absorption lines

Coronal Heating

  • Early observations of the corona revealed a wealth of unidentified lines, and even the supposition that there were new elements in the corona.
  • It was later determined that these lines came from highly ionized species, indicating that the corona is much hotter than layers below it.
  • Suggestions for the heat source include photospheric sound waves and magnetically induced currents.