# 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.