1. Incoherent-Scatter Radar
Theory of ISR presented to the
PARS Summer School 2006

2. Ionospheric Electron Density
The ionospheric plasma parameters and ionic composition varies greatly with altitude. The objective of much ionospheric research is to observe the characteristics of the density, temperature, and composition profiles and how they vary with geophysical conditions.
Thomson scattering

3. Ionosonde
One of the first techniques for observing the electron density profile in the ionosphere was by radio sounding. A radio signal of some given frequency is transmitted and reflects from the ionosphere at an altitude where the local electron plasma frequency is equal to the transmitted frequency.

4. Electron Density

5. Ionogram

6. Ionogram

7. First Incoherent-Scatter Radar
W. E. Gordon of Cornell is credited with the idea for ISR.
“Gordon (1958) has recently pointed out that scattering of radio waves from an ionized gas in thermal equilibrium may be detected by a powerful radar.” (Fejer, 1960)
Gordon proposed the construction of the Arecibo Ionospheric Observatory for this very purpose.

8. Proceedings of the IRE, November 1958

9. First Incoherent-Scatter Radar
K.L. Bowles, Observations of vertical incidence scatter from the ionosphere at 41 Mc/sec. Physical Review Letters 1958:
“The possibility that incoherent scattering from electrons in the ionosphere, vibrating independently, might be observed by radar techniques has apparently been considered by many workers although seldom seriously because of the enormous sensitivity required…”

10. First Incoherent-Scatter Radar
…Gordon (W.E. Gordon from Cornell) recalled this possibility to the writer while remarking that he hoped soon to have a radar sensitive enough to observe electron scatter in addition to various astronomical objects…”
Bowles got the idea for ISR from Gordon and then beat him to the punch. He hooked up a large transmitter to a dipole antenna array in Long Branch Ill., took a few measurements and published.

11. First Incoherent-Scatter Radar
The radar frequency, 41 MHz, was well above the highest plasma frequency in the ionosphere.
Not Total Reflection.
The received power was very small, but was observable at all altitudes where there was significant ionization.

12. Basic Radar Principles
Power/unit area at target
Effective area
Received power
Power/unit area at antenna

13. Basic Radar Principles
Effective area in terms of gain
Received signal to noise ratio
Cross section for volume target
Received SNR for volume target

14. Basic Radar Principles
Effective area in terms of gain
Received signal to noise ratio
Cross section for volume target
Received SNR for volume target

15. Basic Radar Principles
Effective area in terms of gain
Received signal to noise ratio
Cross section for volume target
Received SNR for volume target

16. Basic Radar Principles
Effective area in terms of gain
Received signal to noise ratio
Cross section for volume target
Received SNR for volume target

17. Thomson Scatter
re – Classical electron radius

18. Thomson Scatter
Because scattering is proportional to charge to mass ratio, electrons scatter the energy.
If scattering is by independent electrons:
Scattered power is proportional to the product of the number density and the scattering cross section (Ns0).
Scattered signal is Doppler broadened by electron thermal velocity:
Proton e/m = 9.6x107
Electron e/m = 1.8x1011

19. Thomson Scatter BW = ~600 kHz
Thomson scatter would give two very important ionospheric quantities:
Electron density profile (total cross section)
Electron Temperature profile (Doppler broadening)
Unfortunately, the required receiver bandwidth is large:
430 MHz wavelength is 0.7 m;
assume Te=1500 K:
BW = ~600 kHz

20. Thomson Scatter
Receiver noise power is proportional to receiver bandwidth:
Pn = kTsfB

21. Thomson Scatter Assume reasonable parameters: SNR = 6.2x10-6 G
Pt ~ 2 x 106
N ~ 1x1011
T ~ 300K
R ~ 200 km
DR ~ 10 km
SNR = 6.2x10-6 G
To get SNR = .1, G = 16,130 (= 42dB)

22. Arecibo
Gordon got his radar. The Arecibo Ionospheric Observatory (AIO) was
built by Cornell over the period of 1960 to 1963 for the cost of about $9,000,000 under contract from Air Force Cambridge Research Laboratories funded by the Advanced Research Projects Agency (ARPA).

23. Arecibo 1000’ Diameter Spherical Reflector
62 dB Gain
430 MHz line feed 500’ above dish
Gregorian feed
Steerable by moving feed.

24. First Incoherent Scatter
“…Bowles (1958, 1959) finds that if a powerful radar is beamed vertically upwards, a weak noiselike return is observed from ionospheric heights. Bowles’ observations show that the amount of power contained in these backscattered waves is in approximate agreement with Gordon’s predictions but that the power is spread over a much narrower frequency band that Gordon predicted.” (Fejer, 1960)

25. First Incoherent Scatter
Back to Bowles [1958]: “…When observing a volume deep in wavelengths containing many particles having mutual spacing shallow in wavelengths, the particles can no longer be considered to scatter independently. However, statistical fluctuations of the density of particles on a scale comparable to a wavelength gives rise to a different form of scattering…”

26. Plasma: Not free electrons
Electrons in the ionosphere are not a gas of independent particles. They are one of the constituents of a plasma.
When probing a plasma with a wavelength that is longer than the Debye length, collective effects must be accounted for.
In the ionosphere
n ~ 1e10 – 1e12
Te ~
lD ~ 1-30 mm

27. Collective Effects Results are:
The scattered power is still close to that predicted by Thompson scatter.
Return signal is as if scattered from particles with total cross section of electrons but with mass of ions!
Doppler broadening is much smaller.
Required bandwidth is smaller
Required antenna gain is smaller

28. A Proper Theory
Scattering of radio waves by an ionized gas in thermal equilibrium. J. A. Fejer, Can. J. Phys. Vol 38, , 1960
Theory of incoherent scatering of radio waves by a plasma. J.P. Dougherty and D. T. Farley, Proc. Roy. Soc. London, A, 259, 79-99, 1960.
Electron density fluctuations in a plasma. E. E. Salpeter, Phys. Rev., 120, , 1960
Density fluctuations in a plasma in a magnetic field, with applications to the ionosphere. Tor Hagfors, J. Geophys. Res., 66, , 1961.
Plasma density fluctuations in a magnetic field. E. E. Salpeter, Phys. Rev., 122, , 1961.
A theory of incoherent scattering of radio waves by a plasma. 3. Scattering in a partially ionized gas. J. P. Dougherty and D. T. Farley, J. Geophys. Res. 68, , 1963.
A theory of incoherent scattering of radio waves by a plasma, 4. The effect of unequal ion and electron temperatures. D. T. Farley, J. Geophys. Res., 71, , 1966.
Theory and practice of ionosphere study by Thomson scatter radar. J. V. Evans, Proc. IEEE, 57, , 1969.

29. Reflection at a Discontinuity

30. Scattered Field
Scattering is from fluctuations of the index of refraction.
For an incident wave of frequency w0(>>wp):
Index of refraction in a plasma
Index fluctuation
Scattered Field: Spatial Fourier component of density fluctuation at 2k0

31. Density Fluctuations
Random thermal fluctuations are present in any system.
In a plasma, there are thermal fluctuations in both the ion gas and the electron gas.
If the probing wavelength is longer than the Debye length, the fluctuations dominate the characteristics of the scatter.

32. Density Fluctuations
Thermal fluctuations in an ordinary collision dominated gas can be considered to be made up of sound waves.
In a plasma, the fluctuations are ion-acoustic waves and Langmuir waves.
The probability distributions for the wave modes and their spectrum can be derived by various means.

33. Ion Acoustic Waves Ions Thermal velocity Electron Gas
K = Boltzman’s constant

34. Ion Acoustic Waves
Ions
Thermal velocity
Electron Gas

35. Ion Acoustic Waves
Ion fluid momentum equation (ignoring magnetic field):
Electrons move fast and act to shield the potential. (almost but not quite)
Linearize, assume quasi-neutrality, continuity.
Dispersion relation
wave number (2p/l)
(at l=.75, fia~ Hz)

36. Electron Plasma Waves Electron momentum equation
Ion background does not move to shield charge fluctuations
Same steps but get:

37. Electron Plasma Waves Electron momentum equation
Ion background does not move to shield charge fluctuations
Same steps but get:
Big number
Small correction
~2-10 MHz

38. Wave Spectrum Not to scale Electron Plasma Waves Ion Acoustic Waves
Plasma parameters fluctuate with the waves (density, velocity, etc)

39. Landau Damping
An important detail

40. Landau Damping An important detail Particle distribution function:
f(v) = number of particles with velocity v.

41. Landau Damping An important detail Particle distribution function:
f(v) = number of particles with velocity v.
vf = wave phase velocity (w/k)

42. Landau Damping An important detail Particle distribution function:
f(v) = number of particles with velocity v.
vf = wave phase velocity (w/k)
Particle population modified
Wave damped

43. Damped resonance Waves in a plasma are resonances.
Damped resonances are not sharp
Example – Q of a resonant circuit. fr fr
Damped Resonance
Resonance

44. Wave Spectrum (ISR Spectrum)

45. How about those plasma waves?
wp ~ 2p*(2-10 MHz), k~2p/.75m
Vp~ x106 m/s
Electron thermal velocity ~125,000 m/s
very-very few resonant electrons (no good surfers)
Plasma waves not damped significantly.

46. ISR Spectrum Dependencies We ignored: Number density: Ns0
Ion Acoustic speed:
(Te, Ti, Mi)
We ignored:
Magnetic field
Ion-Neutral collisions
Coulomb collisions
Probably more stuff

47. Cross Section Dougherty and Farley (ion line neglecting collisions)
a is angle from B
h is Debye length
y is plasma admittance function
m is the ratio Te/Ti

48. Cross Section
ISR spectrum as stolen from Millstone web page

49. Cross Section
ISR Spectra stolen from Arecibo Web page

50. Total Cross Section
(For Te=Ti, s=1/2 predicted by Thomson scatter)

51. ISR Measurements Electron density (calibrated)
Electron density (absolute; Plasma line)
Electron temperature
Ion mass/temperature
Bulk plasma line of sight velocity

52. ISR Inferred Parameters
Ion temperature profile
Ion composition
Electric field
Neutral wind velocity
Currents (?)
etc……

53. Conclusion
ISR is an extremely powerful measurement technique based upon a rich plasma physics theory.