1. ESS 154/200C Lecture 2 Upper Atmosphere/Ionosphere

2. Date Day Topic Instructor Due
ESS 200C Space Plasma Physics ESS 154 Solar Terrestrial Physics M/W/F 10:00 – 11:15 AM Geology Instructors: C.T. Russell (Tel. x-53188; Office: Slichter 6869) R.J. Strangeway (Tel. x-66247; Office: Slichter 6869)
Date Day Topic Instructor Due
1/4 M A Brief History of Solar Terrestrial Physics CTR
1/6 W Upper Atmosphere / Ionosphere CTR
1/8 F The Sun: Core to Chromosphere CTR
1/11 M The Corona, Solar Cycle, Solar Activity
Coronal Mass Ejections, and Flares CTR PS1
1/13 W The Solar Wind and Heliosphere, Part 1 CTR
1/15 F The Solar Wind and Heliosphere, Part 2 CTR
1/20 W Physics of Plasmas RJS PS2
1/22 F MHD including Waves RJS
1/25 M Solar Wind Interactions: Magnetized Planets YM PS3
1/27 W Solar Wind Interactions: Unmagnetized Planets YM
1/29 F Collisionless Shocks CTR
2/1 M Mid-Term PS4
2/3 W Solar Wind Magnetosphere Coupling I CTR
2/5 F Solar Wind Magnetosphere Coupling II;
The Inner Magnetosphere I CTR
2/8 M The Inner Magnetosphere II CTR PS5
2/10 W Planetary Magnetospheres CTR
2/12 F The Auroral Ionosphere RJS
2/17 W Waves in Plasmas 1 RJS PS6
2/19 F Waves in Plasmas 2 RJS
2/26 F Review CTR/RJS PS7
2/29 M Final

3. The Gravitational Field
Kepler’s laws of planetary motion
Planets move in ellipses with the Sun at a focus.
Planet-Sun line sweeps out equal areas in equal times.
Square of period proportional to cube of semi-major axis.
Newton’s universal law of gravity
Potential Energy
Kinetic Energy

4. Physics of Collisional Gases
Particles have cross sections
Average separation
Particle flux
Time variation due to
spatial variation
In 3 dimensions,
the continuity equation

5. Thermal Pressure
Thermal pressure (internal, static, scalar) is the mean net transport of momentum through a surface per square meter per second produced by the thermal motion of the gas.
Assume that all particles have the same speed c and 1/6 of the particles move in each of the 6 directions.
The particles that can contribute to momentum transfer from volume 1 to 2 are in the x range c ∆t.
The momentum transfer from 1 to 2 is Φ1,2 is 1/6 (c ∆t A) n (c m)/A ∆t =1/6 nmc2
The momentum transfer from 2 to 1 is Φ 2,1 = – 1/6 nmc2
The net momentum transfer is
Φ1,2 – Φ2,1= 1/3 nmc2
Note dynamic pressure involves non-random velocity and is nmu2 where u is the bulk velocity.

6. Altitude Variation: Troposphere
Pressure falls off with altitude. This gradient in pressure just balances the force of gravity on that volume element of the atmosphere or
dp=mgn(h)dh, if m (mass of atom) and g are constant
, if T constant
n = no exp (-h/H), where H = kT/mg
But equation of state, pρ-γ = constant, links T and n through T = constant nγ-1
At high altitudes, the temperature stops falling.

7. Altitude Variation: Lower Atmosphere
As we go further upward in the lower atmosphere, the temperature rises again. This region is the stratosphere. The boundary between the troposphere and the stratosphere is called the tropopause
The top of the stratosphere is the stratopause. Above this point the temperature again falls. Clearly there is heating at the stratopause.
The region of temperature decrease is called the mesosphere. It ends at the mesopause where the atmosphere again begins to warm reaching about 1000 K at 300 km altitude
The heat for the thermopshere comes from the magnetosphere. Planets without magnetospheres do not have a thermosphere
Below 100 km altitude the atmosphere is well mixed or homogeneous because the atmosphere is very dynamic. Above 100 km the atmosphere becomes heterogeneous.

8. Atmospheric Nomenclature
The atmosphere can be divided into many different zones according to its various properties.
Temperature: Affected by heating, radiation, convection and conduction.
Chemical composition: Homogeneity maintained by mixing.
Vertical transport: By convection, diffusion, ballistic trajectories.
Gravitational binding: Collisional versus collisionless motion
Ionization: Density and temperature vary.

9. The Structure of the Heterosphere
Above about 100km the diffusion of particle species through each other becomes faster than eddy diffusion due to parcels of air interchanging.
Soon the scale height becomes inversely proportional to the mass of each of the species. 40Ar falls off faster than 32O2 that falls off faster than 28N2 that falls off faster than 4He.
Note that the neutral density greatly exceeds the ionospheric density until very high altitudes.

10. Collision Frequency in the Heterosphere
Cross section
Area covered by
targets encountered in
depth l is
1/l nn = s
The mean free path of a particle (l) is (σnn)-1 where σ is the cross section and nn is the number density.
The collision frequency ν1,2 is the thermal speed divided by the mean free path, vth σ nn, where vth is measured in the reference frame of the gas.
The collision frequency drops rapidly with altitude changing from about 1010 times per second at the surface of the Earth to once per second at 400 km.
The mean free path (l1,2 here) varies from 0.1 microns to 100 km over the same distance.

11. Distribution Function
If we wish to calculate the escape flux we need only to know the speed distribution independent of direction
We integrate over all directions and obtain a normalized speed distribution function
The most probable speed is (2 kT/m)1/2; mean speed, (8 kT/πm)1/2 and rms speed, (3 kT/m)1/2
For 1000K H atoms at the exobase, there are some escaping atoms
In the fluid approximation, we use a distribution function to describe the spatial variation (in configuration space) and the velocity distribution
In its equilibrium state a gas can be described by a maxwellian distribution in velocity space
where

12. Atmospheric Escape Flux
Neutrals in the exosphere consist of :
1) ballistic non-escaping particles 2) orbiting particles
3) escaping particles
4) accreting particles
5) transient particles
To determine if an atmosphere is stable over the life of the planet we must calculate the escape flux over the life of the planet.
We can use the distribution function to calculate the escape flux at the exobase, EB, the so called called Jeans escape flux in terms of the escape parameter X
Φes = n(hEB) cw/(2π1/2) (1+X) (exp (-X))
where X = (ces/cw)2EB = rEB/HEB
where HEB = kT∞/(mgEB)
Note: EB is exobase; es is escape; cw is most probable thermal speed
The escape parameter for hydrogen in the Earth’s atmosphere is about 7 and it escapes in less than 105 years but helium with X = 28 is stable over the lifetime of the Earth

13. Energy from the Sun Photosphere temperature is 5780K.
Temperature first drops and then rises slowly in the chromosphere.
Rapid rise of temperature in the transition region.
Temperature reaches over 106 K in the corona.
UV comes from the upper photosphere; Ca II, Ha, K, H emissions from chromosphere; EUV, X-ray and radio waves (10.7cm) from the corona.
Energy of sun is produced by nuclear fusion.
This energy is transported upward by radiation and then by convection when it becomes more efficient.
Convection makes the Sun's photosphere turbulent, drives the Sun's dynamo and introduces short time scales into solar variations.
Photosphere is the layer from which 50% of the photons can escape the Sun without being absorbed.

14. Solar Radiation Spectrum
Radio emissions at wavelengths produced in coronal near the location of EUV production have large variations that are moderately but not perfectly well correlated with EUV variations.
Upper atmosphere absorbs EUV and part of UV spectrum and is heated significantly by this radiation.
Solar EUV flux can vary by up to a factor of 4.
When ozone in the stratosphere is destroyed, UV radiation can reach the surface of the Earth causing an increased risk of skin cancer.
Most of the radiance from the Sun is in visible wavelengths. That is why we use it to see.
Sun is most variable at X-ray and EUV wavelengths and moderately variable at UV wavelengths.

15. Waves in the Atmosphere
Acoustic waves are pressure fluctuations that travel at the speed of sound vs
As illustrated here pressure gradients accelerate the gas back and forth.
A standing wave would have zero velocity at the peaks and troughs. A traveling wave as illustrated here has gas velocity 90 deg out of phase with pressure forces.
Buoyancy oscillations occur in the vertical direction at the Brunt-Vaisala frequency
If speed of sound or temperature is constant, ωg = g (cpT)-½
If we combine buoyant and compressional effects, we get gravity waves illustrated here.
Waves at frequencies above ωa = γg/2vs are called acoustic gravity waves.
Waves below the Brunt-Vaisala frequency are called internal gravity waves.

16. Photoionization
As radiation passes through the atmosphere, it is absorbed and its intensity decreases
If this absorption is due to ion production, then
One ion pair is produced (generally) per 35eV in air
Production is a maximum when
Here
Since
So peak production occurs when
or where , where Nnm is the integrated density. This is also where the optical depth is unity.

17. Photoionization – Details
Intermediate steps in deriving peak production location:
Rewriting in terms of height (h), rather than path distance (s):
Density at peak production altitude:
To determine integrated column density, rewrite density in terms of s:
Optical depth:

18. Chapman Production Function
Peak production is
Production as a function of height is
Let y = (h - hm) / Hn then
Below the peak y is negative and exp (-y) dominates
Above the peak y is positive and –y dominates
If we reference local production rate to maximum at subsolar point, we obtain
where

19. Particle Impact Ionization
In many situations, particle impacts can be the principal source of ionization
Solar proton events in polar cap
Auroral zone during substorms
Satellites with atmospheres in planetary magnetospheres
A primary particle can produce energetic secondary electrons that can ionize. These electrons can also produce x rays when they decelerate
Charge exchange can occur for ions producing a fast neutral
Process is very non-linear; often is numerically stimulated
Range energy relation is a good approximation. Allow calculation of stopping altitude.
Note: In this context nn(s) is a mass density

20. Particle Energy Deposition
Need to calculate altitude distribution of energy loss
Range-energy relation can also be written
Assume that the depth of matter traversed at x is approximated by
Where is energy particle has at point x
Assume
Solving for
Then
Curves here are for mono-energetic beams. In practice, sum over a distribution of energies.

21. Bremsstrahlung/ Ion Loss
Electrons scatter much more easily than ions.
A decelerating or accelerating electric charge produces electromagnetic energy.
This braking radiation tends to be in the x-ray range and this produces further ionization.
Once produced electrons are lost by three processes:
Radiative recombination
e+x+→x+hυ
Dissociative recombination
e+xy+→x+y
Attachment
e+z→z-

22. Ionospheric Density Profile
Photochemical equilibrium assumes transport is not important so local loss matches local production.
If loss is due to electron-ion collisions, we get a Chapman layer
If there is vertical transport
Treating the pressure forces of electrons and ions and assuming neutrals are stationary, we obtain
Where is the ambipolar diffusion coefficient and Hp the plasma scale height
Vertical transport velocity becomes

23. The Earth’s Ionosphere
The electron density in the ionosphere is less than the neutral density.
For historical reasons, the ionospheric layers are called D, E, F
D layer, produced by x-ray photons, cosmic rays
E layer, near 110 km, produced by UV and solar x-rays
F1 layer, near 170 km, produced by EUV
F2 layer, transport important
Enhanced ionization in the D-region leads to absorption of radio waves passing through because it is collisional with neutrals.
At night, ionosphere can recombine, but transport, especially from high altitudes can be important
In polar regions where field is vertical, a polar wind of light ions can form similar to the solar wind.

24. Day/Night Density Profiles
Density profiles from International Reference Ionosphere
Solid lines – Summer
Dashed lines – Winter
Upper plot – high sunspot number
Lower plot – low sunspot number
Note: Units are #/m3, = 106 x #/cm3
MHz

25. Collision Frequencies
Ion and electrons collide with neutrals as they gyrate. How they move in response to electric fields depends very much on the collision frequency relative to the gyro-frequency.
If the gyro-frequency is much lower than the collision frequency, ions and electrons move in the direction of the electric field or opposite to it. This will produce a current.
If the collision frequency is much lower than the gyro-frequency, ions and electrons drift together perpendicular to the magnetic field.
Since the ions and electrons have different gyro-frequencies and collision frequencies, a complex set of currents may be produced. This is treated with a tensor electrical conductivity.

26. Conductivity Parallel equation of motion
Perpendicular equation of motion
Conductivity tensor
Pedersen conductivity (along E┴)
Hall conductivity (along -E x B)
Parallel conductivity