Corresponding author: D. Oberoi, National Center for Radio Astrophysics, Tata Institute of Fundamental Research, Pune, Maharashtra 411007, India. (email@example.com)
 Recent technological advances have led to a resurgence of interest in low frequency radio astronomy. Ionospheric distortion of cosmic radiation has, however, been a challenge for high fidelity and high sensitivity measurements at these long wavelengths. Several new and innovative low radio frequency interferometers are currently in varying stages of development, construction and commissioning across the globe. They will pursue a broad range of scientific objectives, and precise ionospheric calibration over the wide field-of-view of these new generation instruments will be a prerequisite for achieving these science goals. The task of calibration is made more difficult by the Faraday rotation (FR) of polarized flux as it passes through the magnetized plasma of the ionosphere, the plasmasphere, the magnetosphere, and the heliosphere. To quantify these effects, we present a survey of the order of magnitude of FR associated with these media and their spatial and temporal variations.
 Radio astronomy was born at low frequencies (LF) [Jansky, 1932] and this work laid the foundations for much of modern radio astronomy. The bulk of the effort rapidly shifted to higher frequencies for reasons ranging from: the quest for higher resolution; the technological advances which made working at higher frequencies possible; and the inability of the then available technology and techniques to adequately deal with the propagation effects imposed by passage of cosmic radiation through the dynamic and structured terrestrial ionospheric plasma.
 We now seem to have come a full circle; of all the spectral windows accessible from the Earth's surface, the LF radio window is perhaps the one which has been explored with the least sensitivity and fidelity. The impressive and continuing advances in the computational capacity and affordability of digital technology, accompanied by those in calibration and wide field imaging have led to a resurgence of interest in the LF band. A number of different groups across the globe are currently pursuing different approaches to building the next generation of LF interferometers; the Precision Array to Probe the Epoch of Reionization (PAPER) [Parsons et al., 2010] in South Africa, the Low Frequency Array (LOFAR) [de Vos et al., 2009] in Europe, the Long Wavelength Array (LWA) [Ellingson et al., 2009] in the USA and the Murchison Widefield Array (MWA) [Lonsdale et al., 2009; Tingay et al., 2012] in Australia.
 In order to recover the true cosmic signal the LF arrays need to undothe spatially and temporally variable distortions imposed by the passage through the various plasmas through which this radiation has passed on its way to the array, most notably the terrestrial ionosphere. In general this requires determining and correcting for a time and frequency dependent effect, specific to the line-of-sight (los) of observation and also the location of the array element on the ground, a truly daunting prospect. The number of independent degrees of freedom in the problem can be reduced significantly by exploiting the natural continuities in time, frequency, direction of observation and the location of array elements on the ground, making it potentially tractable. In addition to the usual phase delay and attenuation on passing through a plasma, the linearly polarized component of the cosmic signal undergoes a Faraday rotation (FR) due to the magnetized nature of this plasma. Much of the effort in ionospheric calibration has so far focused on determining the ionospheric phase, refraction and attenuation [e.g.,Jelić et al., 2010]. Lawrence et al.  provide an old but excellent survey of ionospheric propagation effects. The FR component of the calibration problem has yet to receive as much attention, perhaps due to the expectation that very little linear polarization is expected to survive in this part of the spectrum.
 The science objectives of many of the new generation LF arrays, however, include observations which require unprecedented calibration precision, e.g., the studies of the Epoch of Reionization (EOR) with signal strength of order 10 mK in the vicinity of 150 MHz [e.g., Bowman and Rogers, 2010] where the Galactic background strength is about 250 K [Rogers and Bowman, 2008]. While the fraction of linearly polarized flux falls as one proceeds to lower frequencies, it does not drop to zero. The sources of polarized emission at these frequencies include both discrete sources such as the extragalactic continuum sources [e.g., Haverkorn et al., 2003a, 2003c] and pulsars [e.g., Manchester, 1971; Johnston et al., 2008], and the ubiquitous diffuse galactic synchrotron background [e.g., Haverkorn et al., 2003b; Wijnholds et al., 2010].
 The Galactic synchrotron background emission is rather strong at LF and the brightness distribution of its total intensity [e.g., Ko and Kraus, 1956; Haslam et al., 1982; de Oliveira-Costa et al., 2008] and average spectral index [e.g., Costain, 1960; Rogers and Bowman, 2008] are fairly well characterized. There have been far fewer attempts to characterize the polarization of the Galactic background [e.g., Muller, 1963], here we rely on the comparatively recent studies around 350 MHz done using the Westerbork Synthesis Radio Telescope [Wieringa et al., 1993; Haverkorn et al., 2000, 2003a, 2003b, 2003c]. These observations cover more than 100 deg2 on the sky and usually sample galactic latitudes, b, in the range 6°–16°. Their key conclusions are that the linearly polarized component of galactic synchrotron background radiation is (1) ubiquitous, (2) has brightness temperatures in the range 3–13 K, and (3) generally has a mottled structure on arc-minute angular scales, though no corresponding structure is seen in the total intensity maps. Observations at higher galactic latitudes (b = +71°) support similar conclusions, with the brightness temperatures of polarized emission in the range 3–5 K [de Bruyn et al., 2006]. Observations of at least one mid-galactic latitude field (b = 30°) have found that the polarized galactic background at 150 MHz is substantially weaker than expected based on extrapolations from around 350 MHz [Pen et al., 2009], though this field does not have corresponding observations at 350 MHz.
 The magnitude and nature of the expected EOR signal, requiring integration times of order 1000 hours, and in general the expectation, or at least the desire, for the new generation LF instruments to achieve very high dynamic range imaging capabilities, require a careful consideration of possible sources of systematics and errors, and provide the motivation for this work. Here we present a survey and order of magnitude estimates of expected FR due to different magnetized plasmas in the terrestrial neighborhood through which the cosmic signals must pass, along with estimates of their spatial and temporal variability, when possible. These media include the heliosphere, the magnetosphere, the plasmasphere and the ionosphere.
 In Section 2 we establish the basics of FR and rotation measure (RM). Sections 3 and 4 discuss the FR signature of heliospheric and magnetospheric plasmas, respectively. Section 5 discusses Plasmaspheric and Ionospheric FR and their variability is discussed in Section 6. A brief summary is presented in section 7 and the conclusions are presented in Section 8.
 The RM due to the interstellar medium (ISM) can exceed that from the media discussed here by orders of magnitude [e.g., Taylor et al., 2009]. The RM due to ISM, however, does not change on typical observing timescales and, in the present context, we regard it as the astronomical phenomenon under investigation rather than a contaminant. A significant observational consequence ISM of relevance at low radio frequencies is that observations toward directions with high RMs (e.g., low Galactic latitudes) can suffer from bandwidth depolarization. A more detailed discussion of the RM due to ISM is beyond the scope of this work.
2. Rotation Measure and Faraday Rotation
 For a given distribution of electron density, Ne(R), and magnetic field, B(R), the rotation measure (RM), in SI units, is given by:
where e is the charge of an electron; m is the mass of an electron; c is the speed of light; ϵ0 is the permittivity of vacuum; R is the coordinates of a point in a three dimensional space; s is a vector coordinate measured in m along the los through the plasma; “·” represents the dot product between vector quantities; and RM is measured in rad m−2. The limits of integration run along the los from the source of cosmic radiation to the Earth based observer, though only the part of the los with significant Ne(R) and B(R) contribute to the integral. Faraday rotation, the angle through which the plane of the linearly polarized radiation rotates on passage through a given magnetized body of plasma, is given by:
where λ is the wavelength of observation in m. A RM of 1 rad m−2 corresponds to a rotation by 4 rad (∼230°) at 150 MHz, and a phase ramp of ∼13 rad (∼745°) across the 80–300 MHz band.
3. Heliospheric Faraday Rotation
 The heliosphere is defined to be the region around the Sun which is dominated by plasma of solar origin, the solar wind. Close to the Sun, the high coronal density and magnetic field strength lead to an observable coronal FR signature. Observations of background polarized radio sources near solar conjunction have long been used for probing coronal magnetic fields and densities in these regions not accessible to in-situ probes. The source used for background radiation have ranged from spacecraft beacons [e.g., Levy et al., 1969; Bird and Edenhofer, 1990] to extended extra-galactic continuum sources [e.g.,Sakurai and Spangler, 1994; Mancuso and Spangler, 2000; Spangler, 2005] and pulsars [e.g., Smirnova et al., 2009]. Bird  provide a very useful overview of the coronal FR measurements, and Spangler and Whiting  provide an excellent discussion of quantitative estimates for coronal FR, consistent with existing observations, and the information that can be obtained from these observations. Based on these the coronal RM due to the background heliosphere is expected to be ∼6 rad m−2 at an elongation, ϵ, of 2.5° from the Sun, and drops by an order of magnitude to ∼0.5 rad m−2 by ϵ = 5°. Due to the roughly R−2 fall in Ne(R) and an even steeper decline in B(R), the RM signature of the quiescent heliosphere falls very steeply with R. By about ϵ = 5° it is already comparable in magnitude to the ionospheric RM, and is expected to fall below the detection threshold of the instruments being considered here by few tens of degrees.
 As the coronal plasma with its inherent azimuthal and latitudinal structure co-rotates with the Sun it leads to a time variable RM signature. Along with this slow geometric component producing changes over hours and days, additional variability is introduced by the constant evolution of the coronal features, which can be on much faster timescales. Certain coronal structures tend to have large scale alignments of magnetic fields and higher than ambient electron densities, and hence can be expected to give rise to stronger RM signatures. Additionally, the location of the magnetic neutral line along the los has a significant impact on the net RM measured along it; the closer it lies to the point of closest approach to the Sun, the larger is the expected RM [Ingleby et al., 2007]. The coronal structures of most interest are the Corotating Interaction Regions (CIRs) and the Coronal Mass Ejections (CMEs), and are discussed next.
3.1. Corotating Interaction Regions
 CIRs are formed at the interfaces of the fast and slow solar wind region, where the faster solar wind rams into regions of slower solar wind ahead of it [Balogh et al., 1999]. As the fast streams typically arise from long lived coronal holes, these structures are relatively stable and co-rotate with the Sun. They develop into shocks, typically close to or a little beyond 1 AU and have been detected as coherent structures out to tens of AU (1 AU = 1.49 × 1011 m). CIRs are most common in the Ecliptic plane, though are seen at latitudes of up to ∼±45° close to solar maxima. As they move outwards, they sweep up more material and at low heliographic latitudes in the outer heliosphere a large fraction of the mass and magnetic field flux of the solar wind is found within the CIRs.
 To get an upper limit on the RM expected from a CIR, we assume Ne and Blos, the los component of the B, to remain constant along the part of the los threading the CIR. We use generous values of 1 × 107 el m−3 for Ne in the vicinity of 1 AU, a los component of B of ∼5 nT and a path length of order 1 AU [Crooker et al., 1999]. This leads to a RM of ∼0.002 rad m−2, or a FR by ∼0.5° at 150 MHz. The RM signature will peak at a patch in the sky where the los aligns well with the Band will slowly fall off over scales of few tens of degrees. As the CIR co-rotates with the Sun, changes in observing geometry will lead to changes in the observed RM signature, including its apparent motion in the sky at a rate of about 13° day−1. Consideration of alignment of the los with the CIR Archimedian spiral suggests that the patch in the sky where the CIR contribution to RM will be the largest will lie in the night sky.
3.2. Coronal Mass Ejections
 CMEs are large scale expulsions of magnetized plasma into the corona and the solar wind. These are very violent events driven by the sudden release of magnetic energy, and can give rise to observational signatures all the way from high energy X-rays to low radio frequencies. The properties of an average CME are now reasonably well established [e.g.,St. Cyr et al., 1999]. The magnetized CME plasma produces FR signature and it has been measured by using observations of linearly polarized spacecraft beacons near solar conjunctions. Bird et al.  present some excellent examples showing change in RM ranging from a few 10 s to few 100 s of rad m−2 associated with CMEs crossing the los to the spacecraft at solar offset in the range ∼3–8 solar radii. No measurements of FR due to CME plasma are available yet at large solar offsets and one needs to rely on simulations to build quantitative expectations. Jensen et al.  estimate the RM from CMEs to be ∼0.05 rad m−2 (12° at 150 MHz) at about 0.4 AU and Liu et al.  also arrive at a similar estimate. CMEs grow rapidly in size as they travel outwards, often spanning more than 90° across the plane of the sky by the time they are at 0.5 AU. Regions with particularly favorable geometries can lead to stronger RM signatures.
 Both CIRs and CMEs involve shocks involving density contrasts of an order of magnitude or larger and have enhanced turbulence at these interfaces, leading to discernible changes in the local propagation properties of the interplanetary medium. Interplanetary Scintillation (IPS) studies routinely exploit these changes to track and characterize CIRs and CMEs [e.g., Breen et al., 1997; Manoharan, 2010]. A detailed discussion of these effects is beyond the scope of this work.
4. Magnetospheric Faraday Rotation
 The terrestrial magnetosphere is defined to be the region around the Earth dominated by the Earth's magnetic field. Due to its high conductivity, the solar wind is constrained to follow the magnetic lines of field threading it. Unable to penetrate the geomagnetic field, the solar wind sweeps around it and, in the process, severely distorts the magnetosphere. On the sunward side, the pressure balance boundary, referred to magnetopause, is at about 10 REalong the Sun-Earth line [Cahill and Amazeen, 1963], where RE represents the radius of the Earth (1 RE∼ 6,400 km). The magnetosphere extends into a long magnetotail in the anti-sunward direction [Ness, 1969]. On the night side, the geomagnetic field tends to run along the Sun-Earth direction beyond 10RE [Ness, 1969]. It reverses direction at the central (equatorial) plane, referred to as the neutral sheet, pointing toward the Earth in the Northern lobe and away from the Earth in the Southern lobe. The magnetotail is roughly circular and has a diameter of about 30 RE. Its length is uncertain, though it has been detected downwind beyond 1010 m (∼150 RE).
 At nighttime, we will be looking through the tail of the magnetosphere, or close to it, and during solar observations our lines of sight will pass close to the nose of the magnetosphere. It is hence useful to compute order of magnitude estimates for the magnetospheric RM along the anti-solar and solar directions. To estimate the largest nighttime RM, we consider aB(R) aligned with the los close to the center of the magnetotail. We use a path length of 1010 m, a constant Ne(R) of 106 el m−3 [Hargreaves, 1992], and a constant B(R) of 20 nT [Escoubet et al., 1997]. This leads to a RM of 5.2 × 10−5 rad m−2 or a FR of 0.01° at 150 MHz. On the day side, B(R) drops from ∼450 nT at ∼4 RE to ∼125 nT at ∼10 RE [Cahill and Amazeen, 1963] and Ne(R) is in the range 106–107 el m−3. Using mean values for these ranges and assuming the B(R) to be aligned with the los for a path length of 10 RE, leads to a RM of 2.3 × 10−5 rad m−2 or a FR of 0.006° at 150 MHz.
 The magnetosphere is regarded as a violently active region showing very large variability. Even if the variability in its RM signature is an order of magnitude larger than the mean estimated here, it will still be too small to have any practical consequences for the instruments under consideration.
5. Plasmaspheric and Ionospheric Faraday Rotation
 The plasmasphere and the ionosphere are defined to be the regions from ∼1,000–30,000 km and ∼100–1000 km above the Earth's surface, respectively. To be able to estimate their RM contribution, we need quantitative information about the Ne(R) and the B(R) distributions in these regions, which, in turn, depend on the geographic location under consideration. In this work, we use the midlatitude Western Australian location of the MWA as an illustration (geo-magnetic latitude = −38°).
5.1. The Plasmaspheric and Ionospheric Electron Density
 We use the Global Core Plasma (GCP) model [Gallagher et al., 2000] to obtain estimates for the ionospheric and plasmaspheric electron densities. This model uses the International Reference Ionosphere (IRI) [Bilitza, 2001] up to a height of 2,000 km above the Earth's surface and extends the model out to 10,000 km. The GCP model provides electron density as a function of height above the Earth's surface. The total electron content (TEC) column density can be obtained by integrating along the relevant los through the medium. We consider two extrema to determine the expected range of TEC. A winter midnight, during a solar minimum and a period of very low solar activity was used to determine the lower end of the expected TEC range, and a mid-day during spring at solar maximum and a period of high solar emissivity was used to determine the higher end of the expected TEC range. This model provided values of 7.5 and 108.5 TECU, respectively (1 TECU = 1016 el m−2) for the MWA site. Geomagnetic conditions are assumed to be quiet in both the cases. As seen in Figure 1a, the vertical ionospheric Ne(R) profile peaks at heights between 200–500 km, referred to as the F layer. Over the 100–10,000 km range of the ionospheric model, Ne(R) varies by ∼3 orders of magnitude for the low TEC case, and a little less for the high TEC case. We note that during the recent solar minimum, the observed TEC values at this location dropped to about half of their lowest expected value considered here.
5.2. Terrestrial Magnetic Field
 The International Geophysical Reference Field (IGRF) model is thought to provide a reliable description of the Earth's B(R) [Finlay et al., 2010]. The IGRF is a series of mathematical models of the Earth's mean B(R), based on sources internal to the Earth, it accounts for the secular variations, and is updated every five years. We use the prediction from this model at the MWA site. B(R) is predicted to be 55,686 nT at a height of 100 km, falls to ∼62% of its value at 1,000 km and by two order of magnitude at 30,000 km (Figure 1b). At the location of the MWA site, the ratio of the radial component of the magnetic field to the total magnetic field is 85 ± 2% over the entire 100–30,000 km range. We note that the presence of large amounts of sub-surface ferrous deposits can lead to significant differences between the observed and predicted ofB(R).
5.3. Rotation Measure Estimate
 The expected RM for both the low and high TEC cases for a los to local zenith is computed using equation (1). Figure 1c shows the RM contributions of ionospheric and plasmaspheric layers as a function of height for these cases. The total RM, obtained by integrating along the los, for the low and high TEC cases are −0.65 rad m−2 and −8.31 rad m−2, respectively (∼147° and ∼1,916°, respectively at 150 MHz). The magnitude of these numbers underscore the strong influence of ionospheric and plasmaspheric propagation on the appearance of the polarized sky at low radio frequencies. About 90% of the RM comes from the ionosphere, and the remaining 10% from the plasmasphere (Table 1). The Global Positioning System (GPS) satellites orbit at an altitude of ∼20,000 km, about 67% of the way out to the plasmasphere boundary. However the contribution of the plasma above the GPS orbital heights is expected to be of order 0.1%, so GPS TEC measurements capture 99.9% of the expected ionospheric and plasmaspheric RM (Table 1). A useful working number to keep in mind is that 1 TECU of electron column density distributed across the usual height profile leads to ∼23° of FR at 150 MHz.
Table 1. Contributions of Different Slabs in the Ionosphere and the Plasmasphere for the High TEC Casea
Height Range (km)
RM (rad m−2)
Contribution (%) to RM
The slab boundaries are arbitrary, but have been chosen to illustrate the rapid gradation in the RM contribution with increasing height. As mentioned in the text, the contribution from the region above 10,000 km is an overestimate of the true value, but it is such a small fraction of the total RM that it is not a significant source of error.
 While the IGRF B(R) model extends to 30,000 km covering the entire plasmasphere, the Global Core Plasma Ne (R) model only extends out to 10,000 km. For the purpose of computing the RM, we regard Ne(R) in the 10,000–30,000 km range to remain constant at its value at 10,000 km. Though it can lead to a significant overestimate of RM from this region, as shown in Table 1, this constitutes a tiny fraction of the total RM (<0.4%) and does not compromise the validity of the overall estimates.
6. Variability in Plasmaspheric and Ionospheric Faraday Rotation
 We now discuss temporal and spatial variability in the observed RM contribution of the plasmasphere and the ionosphere, and provide a brief survey of the nature of structures commonly found in these regions.
6.1. Variability in Terrestrial Magnetic Field
6.1.1. Physical Mechanisms
 The secular change in B(R) due to sources internal to the Earth is of order few nT year−1. Even though these secular changes are ∼4 orders of magnitude smaller than surface fields, they are believed to be captured well by the IGRF models. Atmospheric currents (including ionospheric, plasmaspheric and magnetospheric currents) give rise to much larger variability in the B(R).
 The ionospheric currents, which arise in response to solar forcing, lead to diurnal variations in the mean B(R) at any given location. These diurnal variations can cause the horizontal component of the local B(R) to vary by 50–100 nT.
 The outer regions of the terrestrial magnetic field are distorted due to the influence of the magnetospheric tail, which always points away from the Sun. This along with the inclination of the Earth's rotation axis with respect to the Ecliptic plane gives rise to a seasonal variation of order few tens of nT in B(R). An exhaustive review by Courtillot and Le Mouel  describes a host of different contributions to temporal variability of geomagnetic field at timescales ranging from 1000 Hz to 100 Myrs, none of them large enough to merit particular attention here. Another known kind of variability in B(R) is referred to as “magnetic pulsations”, quasi-sinusoidal variations of amplitude by 5–10 nT on 5–500 s timescales, are also too weak to be of interest here.
 At the midlatitudes, the variations in the plasmaspheric ring current during geo-magnetic storms is expected to lead to the largest changes inB(R). The ring currents are generally located 4–6 RE above the Earth's surface. The strength of B(R) induced by the ring current is inversely proportional to the distance from the current, so the strength of the perturbation to B(R) increases slowly with R through the ionosphere. By the time it grows to a considerable fraction of the value of the unperturbed B(R) high in the plasmasphere, the Ne(R) has fallen by ∼2 orders of magnitude below its peak. As the plasma in the region 10,000–30,000 km contributes <0.5% of the total ionospheric and plasmaspheric RM, these storms can be expected to produce time variations of a similar magnitude.
6.1.2. Observational Characterization
 At high latitudes, the locally observed variability in B(R) is characterized in terms of the K index [Bartels et al., 1939]. The K index is derived from the maximum fluctuation of horizontal component of B(R), BH(R), during a three hour interval, as measured by magnetometers at ground based observatories. It is a quasi-logarithmic index of geomagnetic activity relative to an estimated undisturbed or regular quiet day variation for the recording site. It is an integer scale ranging from 0–9. The level of disturbance increases with increasing numbers and 5 or more indicates a geomagnetic storm. The Planetary K index, Kp, is used to characterize the global geomagnetic conditions [Hargreaves, 1992]. Kp index is constructed from a weighted average of 13 auroral observatories distributed in longitude across the globe.
 The Disturbance Storm Time geomagnetic index, Dst, which monitors the world wide magnetic storm level [Hargreaves, 1992], is perhaps of more relevance to the LF radio arrays as it quantifies the departure from average conditions at lower latitudes. It is constructed by averaging BH(R), from low-latitude and equatorial magnetogram observatories distributed across the globe. It is expressed in nT and is based on the average value ofBH(R) measured hourly at these observatories. Its use as an index of storm strength is based on the inverse relationship between the strength of the surface magnetic field at low latitudes and the energy content of the ring current, which increases during geomagnetic storms. An increased strength of the ring current leads to an induced magnetic field in a direction opposite to the terrestrial horizontal field, leading to a reduction in the observed value of the magnetic field. The Dst index is corrected to remove the contributions of the quiet time ring current and the magnetopause current. A Dst of −100 nT, is a good representative number for a typical storm, “super storms” can push the Dst to ∼−500 nT. The “main” or the build-up phase of typical storms can last for 1–3 days and the “recovery” to normal Dst levels can take 3–5 days.
6.1.3. Spatial Variability in Observed FR Due to Geometric Effects
 Even if the ionospheric and plasmaspheric B(R) and Ne(R) were to be exactly constant, due to the geometric effects involving the differing lengths of the los through these media and the variation in alignment with the terrestrial magnetic field, the observed FR will show a spatial structure. For instance, at the MWA site, the IGRF model predicts the dip angle, the angle the magnetic field makes with the horizontal, to be ∼−60° and the magnetic declination, the angle between the magnetic north and the true north is <1°. Hence the los with the largest ionospheric and plasmaspheric FR will be the one with azimuth (az) ∼0° and elevation (el) ∼60°. The los to zenith, where the path length through the plasma will be ∼15% shorter and the los component of magnetic field ∼18% weaker, will see about 30% less RM than the los toward az = 0°, el = 60°. The los toward az = 180°, el = 60° can similarly be estimated to see about 50% of the peak RM seen. These geometric effects can lead to RM changes by up to factors of about 3. If there are no variations, these geometric effects approximately repeat daily.
6.1.4. Assessment of Impact on LF Observations
 Based on information presented here, for the purposes of estimating their RM contribution, the LF arrays should be able to treat B(R) as a constant, unperturbed by the currents in the upper atmosphere, at least during the quiet geo-magnetic conditions (K < 3). Even a K = 5 geo-magnetic storm corresponds to a fluctuation inBH(R) in the range 70–120 nT, only ∼0.4% of the total BH(R) at the MWA site. The low magnitude of this number suggests that RM changes due to variations in BH(R) will not pose much of a challenge to the LF arrays being considered here.
6.2. Variability in Ionospheric and Plasmaspheric Electron Density
6.2.1. Sources of Information
 The information about variability in the ionosphere and plasmasphere presented here comes from the Los Alamos VLBI array and GPS observations.
 From 1993 to 1997 the Los Alamos VLBI array was used to routinely observe ionospheric and inner plasmaspheric disturbances by measuring the phases of the beacon signals from multiple satellites which backlit the plasmasphere and ionosphere from a geo-stationary orbit at about 35,000 km altitudes [Kirkland and Jacobson, 1998]. This was a roughly Y shaped array with 9 elements and baselines ranging from ∼15–170 km. These observations offer some unique advantages:
 1. Observations of multiple geo-stationary satellites allowed these observations to distinguish between ionospheric and plasmaspheric variability, and to some extent, between temporal and spatial variability.
 2. Unlike astronomy where the los tracks a celestial source, a fixed los through the ionosphere and plasmasphere made it more useful for characterizing these regions.
 3. The beacon frequency of 137–138 MHz was low enough to be sensitive to small changes in TEC.
 4. The system noise for this setup was at the level of 1013 el m−2, or mTECU levels.
 There are about 30 GPS satellites currently in orbit at altitudes of 20,000 km and periods of 12 hours. At any given time 5–8 of these satellites are usually visible from a given location on the Earth. The GPS TEC measurements are made by ascribing the difference in the arrival time of the signal at L1 (1575.42 MHz) and L2 (1227.60 MHz) to dispersion delay through the ionosphere. The current uncertainties on GPS TEC estimates range from 1–3 TECU. The largest contribution to this error comes from temperature and hardware dependent receiver calibration error, referred to as receiver bias [Rideout and Coster, 2006]. Work is actively being pursued to reduce this error to 0.5 TECU or lower, or about 10° of RM at 150 MHz (A. J. Coster et al., Accuracy of GPS total electron content: GPS receiver bias temperature dependence, submitted to Radio Science, 2012).
 The Los Alamos VLBI array provided very sensitive observations, but at only one geomagnetic location. GPS observations have higher error bars but span a longer time base and provide a global coverage. These two sources of observation therefore complement each other very well.
6.2.2. Variability in Ionospheric Electron Density
 The process of ionization in the ionosphere is driven by the EUV and X-ray solar emission. The ionosphericNe(R) hence shows variability on diurnal, seasonal and solar cycle timescales. In addition there is variability on smaller timescales driven by agents ranging from fluid instabilities to gravity waves. Based on their observational signatures on ionograms and the different physical and chemical processes which dominate them, the ionosphere can be divided into distinct regions. These are designated D, E, F1 and F2 and Table 2, reproduced from Hargreaves , describes their daytime characteristics.
Table 2. Daytime Characteristics of Ionospheric Regionsa
 At nighttime, the electron density in the D and F1 layers drop below detection levels of most instruments, and the E and F2 regions persist but become much weaker. The F region is known to be the most variable, the most anomalous and the most difficult to predict [Hargreaves, 1992]. One of the most remarkable things about the F region is its variability from one day to the next. Most of this variability is thought to be caused by phenomena above the ionosphere and is expected to be related to solar radiation and the solar wind, and a much smaller fraction is expected to be a consequence of activity in the lower atmosphere.
 Below we enumerate a subset of known and comparatively common ionospheric phenomena of relevance to LF arrays which introduce variability with different amplitudes, periods and frequency of occurrence. For estimating the expected RM we assume a typical nighttime TEC value of 10 TECU. A summary of the characteristics, RM and FR due to these phenomena is presented in Table 3.
Table 3. Order of Magnitude Characteristics of Relevant Known Ionospheric Phenomenon and the RM and FR (at 150 MHz) Expected Due to Thema
Velocity (m s−1) and Timescale (s)
Spatial (km) and Angular Scale
Frequency of Occurrence
RM (rad m−2) FR (deg)
The second column provides the observed velocity and timescales of the features, and the third column gives the spatial size of the features along with an angular size, as seen by a terrestrial observer. The typical magnitude of TEC variation is listed in the fourth column and the height at which the phenomena are known to occur is provided in the fifth column. The last column gives the RM, and the corresponding FR angle at 150 MHz, assuming an ionospheric TEC value of 10 TECU.
Medium Scale TIDs
100–300 m s−1
8 × 10−2
reduction in TEC
6 × 10−3
>800 km to
(Kp > 2)
Day-to-day variability Shorter time scale variability
 1. Traveling Ionospheric Disturbances.The coupling of the acoustic and gravity waves in the neutral medium to the ionized component of the medium gives rise to the so-called traveling ionospheric disturbances (TIDs) [Hargreaves, 1992]. Their name derives from the fact that when observed from different sites, these perturbations show a lag consistent with the interpretation of a traveling wave. TIDs are quite common and their occurrence has been reported since the 1950s. They are believed to lie in the F2 layer at altitudes in the range 250–350 km. A variety of different techniques have been used for observing TIDs, they include single dish and interferometric observations of satellite beacons which could track the change in the observed RM [e.g., Hunsucker and Hargreaves, 1988; Jacobson et al., 1995], GPS TEC measurements [e.g., Hernández-Pajares et al., 2006; Kotake et al., 2006; Tsugawa et al., 2007], ionosonde measurements [e.g., Hunsucker and Hargreaves, 1988] and changes in apparent position of cosmic sources due to refractive effects [e.g., Lawrence et al., 1964]. TIDs are known to vary in their periods, size scales, speeds and directions of propagation, and have been classified into two main categories.
 TIDs with spatial scales of order 1000 km, with periods longer than about 30 min and horizontal phase velocities in the range 400–1000 m s−1 are referred to as large scale TIDs (LSTIDs) [e.g., Tsugawa et al., 2004; Hayashi et al., 2010]. They are seen in the midlatitude regions, produce a variation of 5–10% in the observed TEC and are not very frequent, occurring only a few days a month. Medium Scale Traveling Ionospheric Disturbances (MSTIDs), on the other hand are known to occur daily in midlatitude regions [e.g., Tsugawa et al., 2006; Kotake et al., 2007]. They have spatial scales of 100–300 km, show horizontal phase velocities in the range 100–300 m s−1and produce variations of 1–5% of the background TEC. Based on GPS TEC observations from Japan and California, nighttime MSTID activity is stronger in summer and winter, is not correlated with geomagnetic activity and is anti-correlated with solar activity. In Japan, MSTIDs were seen in 25–40% of the 1 hour bins during the summer and winter months and in <15% of the 1 hour bins during spring and autumn. At nighttimes, symmetric patterns are seen between geomagnetically conjugate points. Though these changes seem small, they are large enough to be detectable by the new generation instruments. For a TEC of 10 TECU and a 5% TID strength, the excess TEC due to the TID leads to a density of 5 × 109 el m−3 in the F layer and a FR of ∼10° at 150 MHz.
 2. Equatorial Spread F.Equatorial Spread F (ESF) is so named because rather than coming from a single well defined height representing the peak density of the F layer, under these conditions the radar echoes for the F layer “spread” out over a large range of heights. The ESF activity is usually restricted to ±20° around the magnetic equator and occurs at night between 1900 and 0600 LT, although most instances are pre-midnight. ESF is believed to be caused by low density bubbles which rapidly rise in altitude through the denser background ionosphere. The average density contrast in the bubbles approaches an order of magnitude, though bubbles with density contrasts of 2 or even 3 orders of magnitude do occur. The altitude range is typically 300–500 km, above the F peak, although the larger bubbles can rise up to 1000 km or more during disturbed geomagnetic conditions. The radar backscatter observations show considerable structure on timescales of minutes. The information reported here is based on observations from the Atmospheric Explorer - E satellite [Singh et al., 1997] and observations from the Jicamarca radar in Peru [Hysell and Burcham, 2002]. The ESF bubbles are expected to be of order 50–100 km, are elongated along the direction of the magnetic field [Tsunoda, 1980], and can be a few 100 km in height. In-situ rocket data show that turbulent structures associated with ESF span over seven orders of magnitude in spatial scale, from <0.1 m to >105 m [Kelley et al., 1982]. This turbulence is known to give rise to scintillations in the VHF range [Basu et al., 1978] and even at the GPS L1 frequency [e.g., Valladares et al., 2004]. GPS observations show drops in TEC measurements the range 20–50% to be associated with ESF.
 The size scale and height of ESF bubbles imply that the angular size of these bubbles are smaller than the fields of view of the LF arrays. The scintillation associated with ESF will probably pose a larger calibration challenge to the LF arrays than the change in RM due to plasma density depletion. The equatorial nature of this phenomenon and the mid–high latitude location of the LF arrays implies that, if at all, ESF will be seen only at rather low elevations, or large zenith angles, by these instruments.
 3. Sporadic E. Sporadic E is a propagation anomaly which arises due to the presence of very thin highly ionized regions in the E layer [Hargreaves, 1992]. Rocket and incoherent scatter radar measurements reveal that at midlatitudes, these layers are perhaps less than 1 km in thickness and multiple such layers can simultaneously exist. On ionograms they are seen as echoes at a constant height extending to frequencies higher than is usual for the E layer (∼5 MHz). This phenomenon leads to increasing the range of long distance communication at VHF frequencies to order 1000 km. Most of the work on the subject has been done in context of propagation studies and the presence of Sporadic E is routinely documented and even predicted by the radio HAM community. As the name suggests this is a departure from the normal conditions. Though it can occur at any time, it does show a seasonal dependence, peaking in summer in both the hemispheres. It also has local time dependence, with a peak around the local noon and a smaller peak in late evenings during the summer months. At midlatitudes, these peaks correspond to occurrence for about 50–70% and 30–50% of the time, respectively.
 Being very thin, these layers do not contain much plasma so their FR signature is small. Their density contrast of 1–2 orders of magnitude can however lead to significant refraction and the irregularities within them can lead to scattering. The fact that they can substantially alter the propagation characteristics of the region for the FM band (87.5–108.0 MHz) and sometimes even higher frequencies suggests that they will have a non negligible impact on cosmic radio waves incident on the top side of these layers as well.
 4. Storm Enhanced Densities. Very large variations in TEC and attendant variations in RM can occur during times of Storm Enhanced Densities (SEDs). In a few examples studied in the North American sector [Coster et al., 2003], SED events have shown TEC close to 75 TECU. A large fraction of this plasma (∼50 TECU in the event reported) is at high ionospheric altitudes, somewhere beyond the altitude range probed by the incoherent scatter radars (∼800 km) but below the GPS orbital altitude (∼20 × 103 km). Depending upon its height distribution, this high altitude plasma can contribute RM in the range ∼6–0.6 rad m−2. SED events will be accompanied by refractive scintillation and other severe effects that will prevent useful astronomical observations. Such events are infrequent and should be easy to discern and excise from the data.
 5. Short Timescale Variability. Though models like IRI predict smooth diurnal variations in TEC, the GPS TEC measurements show variations of order 10% even during the quiet Sun times on timescales of an hour or so. It is difficult to decompose the observed variability into spatial and temporal components and it seems to be independent of local time and direction of observation. The observed variation in the peak of the F2 layer, NmF2 [Rishbeth and Mendillo, 2001], as measured by ionosondes can serve as a useful quantitative measure of RM variability. Plasma out to the heights of the F2 layer contributes ∼60% of the total RM, and at night much of the rest of the ionosphere disappears. The “spiky” nature of the observed NmF2 and the time variations of order few tens of percent on timescales of hours suggest accompanying RM variations on similar timescales and of similar magnitudes, especially at night. The measurement error associated with NmF2 is of order 1%, much smaller than the observed variation, implying that the observed variations are indeed real. Furthermore, the period studied by Rishbeth and Mendillo  corresponds to solar minimum conditions (1973–74), these variations might be larger during periods of higher levels of solar activity.
 6. Day to Day Variability.A variety of different factors contribute to the day-to-day variability. Solar forcing and geo-magnetic activity are believed to contribute 80–90% of the observed variabiity and are relatively well understood. This is borne out by the fact that the ionospheric TEC variations have a strong correlation with solar EUV flux and magnetic activity. The most likely origin for the remaining 10–20% of the observed variability is thought to be activity in the lower atmosphere, but is not well understood. A quantitative feel for the level of observed variability can be had from the ionosonde NmF2 data.Rishbeth and Mendillo used data from 13 different ionosonde stations to study the day to day variability during solar minimum conditions. They observed TEC variability of ∼20% during the daytime and ∼33% during the nighttime. Daytime variability is smoothed by the effects of photo-ionization and diffusion, while lack of photo-ionization and the presence of dynamic neutral winds enhance the observed fractional variability at night. The absolute value of the TEC is significantly lower at nighttimes, so while the fractional nighttime variability seems higher, it is lower in absolute units.
 7. Solar Cycle. The EUV output of the Sun changes with the phase of the solar cycle and that produces a corresponding variation in the ionospheric electron content. Though the change in solar irradiance is “global”, the impact it produces on the ionosphere differs across the globe. Huang highlighted this by examining the gradient in observed TEC as a function of sunspot number for different locations, while taking care to remove seasonal variations and isolate diurnal variations. He found that the maximum rate of change of TEC with sunspot number at Sagamore Hill, Massachusetts, USA (geomagnetic latitude ∼54°) (at 1500 local time) is about half of the value obtained in Hawaii (geo-magnetic latitude ∼21°) while the minimum rate (at 0400 local time) is three times larger than the value obtained in Hawaii. The small, slow and smooth changes associated with the solar cycle will not pose significant additional complications for LF arrays.
6.2.3. Variability in Plasmaspheric Electron Density
 The plasmaspheric variability seen is characterized by events with timescales of order 100 s, velocities of order 1000 m s−1and TEC amplitudes in the 0.001–0.1 TECU range. The underlying phenomenon is believed to be motion of the field aligned plasmaspheric structures convecting past the los to geo-stationary satellites. These irregularities are believed to be at heights of about 16,000–19,200 km. In about 26 months of observationsJacobson et al.  saw 111,000 “acceptable wavelike perturbations”. They do not report any seasonal dependence for these disturbances, though their frequency of occurrence has a well defined distribution in local time. It peaks at midnight (0.65 hr−1) and has a broad minimum during 0600–1800 local time window (0.03 hr−1). At the altitudes where these features are believed to occur, the B field has fallen to about 1000 nT, a factor of ∼50 lower than the F layer values. For the larger of these fluctuations, the expected RM is of order 2.6 × 10−4 rad m−2 (FR ∼0.06° at 150 MHz).
 The behavior of these features is known to change with geo-magnetic activity [Hoogeveen and Jacobson, 1997]. They appear more often, convect faster, have larger magnitudes and appear at lower altitudes during times of increased geomagnetic activity. Despite this, at the sensitivity levels of the arrays under present consideration, the plasmaspheric FR is not expected to be of much concern. Refraction and scattering due to the small scale features with large gradients in this region might become discernible at high resolutions and would then need to be calibrated.
 We summarize the expected magnitude and variability in FR due the various media considered here in Figure 2. The numbers for FR provided in the following text are at the reference frequency of 150 MHz. The RM due to the quiescent heliospheric plasma falls very rapidly with increasing solar offset. At low radio frequencies it is expected to remain detectable to the new generation LF instruments out to elongations of order 10°, by when it falls to a few degrees of phase. Again, though it falls rapidly with increasing solar offset, CMEs can lead to measurable transient RM signature over a large fraction of the inner heliosphere, falling to order ten degrees by 0.4 AU from tens of thousands of degrees close the Sun. The RM signature of CIRs is smaller, order a degree in the anti-sunward direction. The magnetospheric FR is expected to be smaller still, less than 0.01°, and in spite of its large expected variability, is too small to be of any practical consequence for LF radio instruments being considered here. Of the media considered here, the ionosphere and the plasmasphere are the most pertinent from radio astronomical FR calibration perspective. They produce FR ranging from hundreds to thousands of degrees of E-vector rotation. The ionosphere contributes about 90% of this FR with the remaining 10% coming from the plasmasphere. In addition to the slow diurnal variation, a wide variety of ionospheric and plasmaspheric phenomena give rise to a range of temporal and spatial variabilities. These variabilities are summarized inTable 3 and span large ranges – from minutes to solar cycles in time and few to thousands of km spatially. They can lead to RMs ranging from a few degrees to order a thousand degrees, and can sometimes also have steep spatial gradients associated with them.
 Though the fractional polarization of cosmic radio sources drops rapidly at low radio frequencies, the surviving linearly polarized signal is large when compared to the sensitivity of the new generation of LF instruments and the expected strength of some of the signatures they aspire to characterize, e.g., EOR. These instruments will be able to discern the FR of this polarized flux due to propagation through the various intervening media, especially the ionosphere and the plasmasphere. In fact, measurement of some of these signals, e.g., the heliospheric signals, forms a part of the science case for some of these instruments [Salah et al., 2005]. Unless adequately calibrated, the contamination from these time and direction dependent effects will compromise the sensitivity and the fidelity of the observations from these instruments. Achieving the science objectives of the new generation LF instruments will hence require appropriate characterization and calibration of FR contributed by the various intervening media.
 It is a pleasure to acknowledge the very helpful and illuminating discussions with Anthea Coster, Philip Erickson, John Foster and Shunrong Zhang, all at MIT Haystack Observatory, and their guidance through this work. We thank the anonymous referees for their constructive suggestions which have helped improve the presentation and clarity of this work. This work was supported by grants from the National Science Foundation Divisions of Atmospheric and Geospace Sciences and Astronomical Sciences to the MIT Haystack Observatory.