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

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

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.

Electron Density

Ionogram

Ionogram

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.

Proceedings of the IRE, November 1958

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…”

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.

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.

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

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

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

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

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

Thomson Scatter re – Classical electron radius

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

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

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

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)

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).

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

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)

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…”

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 ~ 300 - 2000 lD ~ 1-30 mm

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

A Proper Theory Scattering of radio waves by an ionized gas in thermal equilibrium. J. A. Fejer, Can. J. Phys. Vol 38, 1114-1133, 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, 1528-1535, 1960 Density fluctuations in a plasma in a magnetic field, with applications to the ionosphere. Tor Hagfors, J. Geophys. Res., 66, 1699-1712, 1961. Plasma density fluctuations in a magnetic field. E. E. Salpeter, Phys. Rev., 122, 1663-1674, 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, 5473-5486, 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, 4091-4098, 1966. Theory and practice of ionosphere study by Thomson scatter radar. J. V. Evans, Proc. IEEE, 57, 496-530, 1969.

Reflection at a Discontinuity

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

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.

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.

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

Ion Acoustic Waves Ions Thermal velocity Electron Gas

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~1500-2000 Hz)

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

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

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

Landau Damping An important detail

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

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

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

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

Wave Spectrum (ISR Spectrum)

How about those plasma waves? wp ~ 2p*(2-10 MHz), k~2p/.75m Vp~1.5 - 7.5x106 m/s Electron thermal velocity ~125,000 m/s very-very few resonant electrons (no good surfers) Plasma waves not damped significantly.

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

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

Cross Section ISR spectrum as stolen from Millstone web page

Cross Section ISR Spectra stolen from Arecibo Web page

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

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

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

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