Astronomy 217

 
 

Prof. Andrew W. Steiner

 
 
Nov. 1 2021
 
 

TA James Ternullo

Last Time

  • The Structure of Earth

Today

  • Earth atmosphere and magnetosphere
  • Remind Andrew to turn on recording now...

Earth's Composition

  • The coalescing Earth managed to capture some gases from the gas-rich protoplanetary disk.
  • Thus the original or primeval atmosphere was rich in hydrogen and helium, but these rapidly escaped the Earth’s gravity. More complex molecules like methane and ammonia were photo-dissociated by sunlight.
  • Secondary atmosphere formed from outgassing, as gas was released from the heating and differentiation of the mantle and core.
  • This was predominantly carbon dioxide, water and nitrogen.
  • Reactions changed the atmosphere!

Atmosphere Chemistry

  • The composition of the atmosphere has undergone many changes driven by chemistry, both internal to the atmosphere and with the crust and ocean.
  • CO2 dissolved in the early ocean, then reacted with minerals to form carbonates, like CaCO3.
  • With the appearance of algae about three billion years ago, CO2 was further removed from the atmosphere, replaced by O2 which is continually being replenished.
  • In modern times, human production of CFCs (Chlorofluorocarbons) damaged the ozone layer, creating an ozone hole.

Greenhouse

  • The composition of the Earth’s atmosphere can have a large impact on the surface conditions.
  • Gases like carbon dioxide, water vapor & methane have high opacities in the infrared, but lower opacities in the optical.
  • Thus the solar photons with an characteristic temperature of 5800 K (\( λ_{\mathrm{peak}} \approx 500~\mathrm{nm} \)) can pass through the atmosphere.
  • Thermal photons from the Earth, with a characteristic temperature of 290 K (\( λ_{\mathrm{peak}} \approx 10~\mathrm{μm} \)) can not escape as easily.

Global Warming

  • In addition to O2, the other product of photosynthesis is hydrocarbons in the form of biomass.
  • A significant amount of this biomass was sequestered by geologic processes, producing fossil fuels.
  • Modern society, by burning fossil fuels, has reversed this sequestration, increasing CO2 levels in the atmosphere.
  • A corresponding increase in global average temperature has been observed, changing the climate.

Global Consequences

  • Climatologists have modeled the consequences of increased greenhouse gas and the resulting global warming.
  • These consequences include
    • Rise in sea level, due to polar ice melting
    • More severe weather, due to increased atmospheric energy
    • Crop failures due to climate zones changing.
    • Expansion of equatorial deserts
    • Spread of tropical diseases away from the tropics
    • Enhanced habitability of the near polar regions

Hydrostatic Equilibrirum

  • As with the structure of the Sun, atmosphere is determined by the thermal pressure $$ F_{\mathrm{pres}} = P A - ( P + \Delta P) A \quad ; \quad F_{\mathrm{grav}} = - \frac{G M(r) (\rho A \Delta r)} {r^2} $$
  • Setting these two forces equal yields $$ \Delta P A = - \frac{G M(r) (\rho A \Delta r)}{r^2} $$ and this yields a pressure difference as a function of \( \Delta r\) $$ \frac{\Delta P}{\Delta r} = - \frac{G M(r) \rho}{r^2} \quad \Rightarrow \quad \frac{\Delta P}{\Delta r} = - \frac{G M(r) \rho}{r^2} $$

Scale Height

  • For planetary atmospheres, the gases obey the ideal gas law $$ P = n k T = \frac{\rho}{\mu m_p} k T $$
  • The equation of Hydrostatic equilibrium is $$ \frac{dP}{dr} = - \frac{G M(r)}{r^2} \frac{\mu m_p}{k T} P \quad \mathrm{or} \quad \frac{dP}{P} = - \frac{G M(r)}{r^2} \frac{\mu m_p}{k T} dr $$
  • Over a small range in radius, the gravitational acceleration is nearly constant, thus $$ \frac{dP}{P} = - \frac{g \mu m_p}{k T} dr \quad \mathrm{where} \quad g = \frac{G M(r)}{r^2} $$
  • The solution is \( \ln P = r/H + C \) or \( P(r) = P_{\mathrm{surface}} \exp [ - (r-R_{\oplus}/H] \) where \( H = k T/(g \mu m_p) \approx 8~\mathrm{km} \)

Earth's Atmosphere

  • The Earth’s atmosphere is not isothermal, but reveals a complex temperature structure (blue curve) determined by heat transport and local heat deposition.
  • The temperature and density in the stratosphere favor the production of ozone (O3) which shields the surface from UV.
  • The upper part of the thermosphere, the ionosphere, is fully ionized by solar radiation.

Heat Transport

  • The dominant source of heat for the Earth’s atmosphere is the Earth’s surface. The Sun-warmed surface re-radiates infrared thermal radiation, which is absorbed by the atmosphere.
  • In the troposphere, this causes higher temperatures near the surface, leading to convective instability.
  • In the troposphere, this causes higher temperatures near the surface, leading to convective instability.
  • The stable temperature structure of the stratosphere is caused by heating when UV light dissociates O3.
  • Similarly, the thermosphere is heated by ionization.

Scattering

  • Even when molecules are unable to absorb photons of a given wavelength, they can still change the direction or scatter the light.
  • The strength of the scattering depends on the relative size of the wavelength of the light (λ) to the size of the scatterer (L).
  • When \( L \ll \lambda \), Rayleigh scattering governs, thus \( \sigma \propto \lambda^{-4} \)
  • For \( L \approx \lambda \), scattering obeys a \( \sigma \propto \lambda^{-1} \) relation
  • For \( L \gg \lambda \), all wavelengths scatter equally
  • In the atmosphere, scattering can occur on molecules (\( L \approx 0.3~\mathrm{nm} \)), dust particles (\( L \approx 1~\mathrm{\mu m} \)), water droplets (\( L \approx 10~\mathrm{\mu m} \)), and ice crystals (\( L \approx 0.1~\mathrm{mm} \))

Why is the Sky Blue?

  • Sunlight passing through clear sky is scattered by dust and molecules. For visible light (λ =400-700 nm), Rayleigh scattering causes more scattering of shorter wavelengths.
  • Along the light of sight to the Sun, blue light is removed leaving red sunsets and sunrises.
  • The scattered blue light dominates along other lines of sight.
  • In contrast, larger water droplets scatter all wavelengths equally, thus appearing white.

Rayleigh Scattering

  • For light with wavelength \(\lambda \gg L \), the particle size, it is appropriate to consider light as an electromagnetic wave. This moving wave generates movement in the electrons present in matter that scatter or deflect the light. $$ I = \frac{I_0}{r^2} \frac{8 N \pi^4 \alpha^2} {\lambda^4} ( 1 + \cos^2 \theta) $$
  • The strength of the response to these r movements (α, the polarizability) depends on the scattering molecule.
  • Shorter wavelengths (higher frequencies) are closer to the molecules’ natural frequencies, thus strength of the scattering response is stronger, e.g. \( 700^4/400^4= 9.4 \) .
  • Intensity depends on scattering angle θ and distance r.

Magnetosphere

  • Above the ionosphere and the exosphere lies a region protected from the solar wind by the Earth’s magnetic field.
  • On the sunward side, the magnetosphere is compacted by the solar wind, but on the downwind side it extends much further, forming a teardrop shape.

Magnetic Dipole

  • The Earth’s magnetic field resembles a bar magnet. $$ B(r) = B_{\oplus} \frac{R_{\oplus}^3}{r^3} $$
  • Deflecting the solar wind requires a magnetic energy density greater than the solar wind’s kinetic energy. $$ \frac{B^2}{2 \mu_0} = \frac{\rho v^2}{2} = \frac{B_{\oplus}^2}{2 \mu_0} \left( \frac{R_{\oplus}}{r} \right)^6 \Rightarrow \frac{r}{R_{\oplus}} = \left( \frac{B_{\oplus}^2}{2 \mu_0} \frac{2}{\rho v^2} \right)^{1/6} $$
  • For \( B_{\oplus} = 3.1 \times 10^{-5}~\mathrm{T} \) and \( \rho v^2/2 \approx 10^{-9}~\mathrm{J}~\mathrm{m}^{-3} \) the radius of the magnetopause is \( \approx 8.5~R_{\oplus} = 5.4 \times 10^{4}~\mathrm{km} \)

Trapped Particles

  • Solar wind particles that do leak through the magnetopause can become trapped in the Earth’s magnetosphere.
  • The trapping occurs because the particles spiral around the magnetic field lines.
  • The 2 principle regions the particles become trapped are called the Van Allen belts.
  • These belts extend from the top of the Earth’s atmosphere to the magnetopause but are strongest at \( \approx 0.5~R_{\oplus} \) and \( \approx 3.0~R_{\oplus} \)

Aurorae

  • Near the poles, the Van Allen belts intersect the atmosphere. The charged particles collide with molecules and atoms in the atmosphere, collisionally exciting them.
  • The resulting photo-deexcitation produces the glowing light called aurorae, borealis near the north pole and australis in the south.

Magnetic Changes

  • Though the Earth’s magnetic poles lie close to its rotational poles, the alignment is not exact.
  • The magnetic poles move as much as 60 km/year.
  • The new crust created at rift zones preserves the magnetic field present at the time it solidified. From this, we can tell that the magnetic field reverses polarity periodically.
  • Recently, this has occurred about every 500,000 years.