The Low-Frequency Array
3.3 The lowest frequencies
3.3.2 Ionosphere
The ionosphere is a layer of the atmosphere which starts at∼ 100 km above the Earth surface and terminates at∼ 1000 km (see Fig. 3.9). This layer of the atmosphere is highly ionised (ne = 104−106electrons cm−3) and the ionization is due to the interaction between solar radiation and cosmic rays with the Earth’s atmosphere. Clearly, the elevation of the Sun and the Sun activity influence the intensity of the solar radiation which in turn affect the concentrations of free electrons in the ionosphere. Therefore, the properties of the ionosphere show daily, seasonal and decadal variations. Furthermore, they are also dependent on the geographic latitude and on
3.3 The lowest frequencies 63
(a) Time-frequency plot before flagging (b) Time-frequency plot after flagging Figure 3.8: Flagging results on a sub-band around 156 MHz for a 1.5 km baseline of a 6 hour LOFAR observation, from Offringa et al. (2010b). Horizontal axis is time and vertical axis is frequency. In the right image yellow pixels are flagged.
the geomagnetic field strength6.
In the ionosphere the propagation of radio waves is modified by the presence of free elec-trons. In fact, when a radio wave reaches the ionosphere, the electric field in the wave forces the electrons to oscillate at the same frequency as the radio wave. Some of the radio-frequency energy is then given up to this resonant oscillation. The oscillating electrons will then either be lost to recombination or will re-radiate the original wave energy. At frequencies below ' 300 MHz, the ionosphere affects radio waves via (i) refraction, (ii) absorption, (iii) Faraday rotation and (iv) total reflection (for a detailed explanation see Intema et al. 2009 and Sec. 13 of Thompson et al. 2001).
The refractive index for an incident wave of frequencyνon a plasma is n'
s
1− 4πe2ne
meν2 = r
1− νp
ν 2
(3.29) wheree is the electron charge, me is the electron mass, ne is the free electron density andνp
is called theplasma frequencyand in the ionosphere is about 1−10 MHz for typical electron densities of 104−106electrons cm−3. Whenν < νp, the refractive index becomes imaginary and there is total reflection and it is impossible for a telescope to observe. It is interesting to note thatn∝ν−1, therefore the ionospheric effect is more and more important at lower frequencies.
Furthermore, the refractive index is function of the electron density ne which in turn is not uniform across the wide field of view of LOFAR. To summarise, the effect of refraction is the introduction of time (i.e. phase) delays (due to the refractive index being,1) that are direction dependant.
6And up-to-date status of the ionosphere total electron content (TEC) obtained by mapping GPS observables collected from ground stations is available on http://iono.jpl.nasa.gov/latest_rti_global.html.
Figure 3.9: Relationship of the atmosphere and ionosphere. Solid line is the temperature [K] while dashed line is the electron density [cm−3]
Ionospheric absorption instead is a relatively small amplitude effect (e.g. 0.1 dB at 100 MHz andZA =60◦, Thompson et al. 2001, Sec. 13), and is mostly absorbed in the telescope overall gain calibration. Differential absorption are also present and they are a non-trivial DDE, but it is a smaller effect.
Faraday rotation is due to the interaction between light and a magnetic field in a medium and causes a rotation of polarization plane which is proportional to the component of the magnetic field in the direction of propagation. Quantitatively, the angle variation is
∆Φ =2.6×10−17ν−2Z
LoS
neB||dl= RMν−2[rad], (3.30) whereν [Hz] is the frequency, ne [cm−3] is the electron density, B|| [µG] is the magnetic field along the line of sight and dl[m] is the path length along the line of sight (LoS). RM is called therotation measureand provides information on the medium crossed by the radiation during its journey from the source to the telescope. In the RIME formalisms (in a linear-coordinate polarization basis) the Faraday rotation is a rotation matrix of the form
F= Rotβ=
cosβ −sinβ sinβ cosβ
, β∝ ν−2Z
LoS
neBkdl, (3.31) while in a circular basis is a diagonal matrix which encode a simple phase delay. Therefore, the Faraday effect causes left and right circularly polarized waves to propagate at slightly different speeds. In an instrument like LOFAR which has linear polarization receivers this effect moves
3.3 The lowest frequencies 65 some flux of the diagonal terms in Eq. 3.8 to the off-diagonal terms. This is the reason why, when long baselines are used, and thus the crossed ionosphere is particularly different for two antennas, it is common practice to convert the data from linear to circular polarisation, where this effect is a simple delay that can be compensated with a standard calibration.
Finally, ionospheric scintillation is another effect that can alter the propagation of a radio signal. They are the consequence of small-scale variations in the ionospheric electron density ne, and generate diffraction and scattering in the propagating rays. Ionospheric scintillations are characterized by random temporal fluctuations both in amplitude and in phase of a radio signal and can happen both during day or night-time.
The Lonsdale scenarios
In Lonsdale (2004) three ionospheric calibration scenarios have been identified. The scenarios compare the array aperture (the length of the largest baseline) and the FoV with the ionosphere isoplanatic patch size, i.e. the scale at which the ionosphere can be considered constant. Lons-dale (2004) work has been revisited and adapted for LOFAR by Wijnholds et al. (2010).
Scenario 1 (Fig. 3.10a) in this scenario the baselines lengths are short and the FoV of each station is small compared to the size of the isoplanatic patches. In this situation all the receivers and all the lines of sight within the FOV experience the same propagation conditions. We do not have direction dependant effects and the term7 E in Eq. 3.21 is a scalar. Since the FoV is small standard calibration can be applied using single strong point sources to solve for theGterms.
Scenario 2 (Fig. 3.10b) in this scenario the array is large but the FoV is still small. Lines of sight from different elements towards the region of interest are subject to different propagation conditions but within the FoV of a single element all lines-of-sight are ex-periencing the same conditions. The propagation effects can therefore be merged with the unknown receiver gains of each element, this means thatEdoes not depend on (l,m) and can be taken out of the integral in Eq. 3.21 and merged withG. The calibration pro-cedure is then the same of the previous scenario. This scenario is valid for most of the interferometers built in the 70s and 80s, such as the WSRT and the VLA, and for VLBI.
Scenario 3 (Fig. 3.10c) here the elements have a large FOV, but the array is small. Therefore, all lines of sight go through the same propagation path, but there could be differences in propagation conditions towards distinct sources within the FoV. The ionosphere thus
7I use here theE term as a general DDE effect, while usually the “E” letter is connected to the beam effect.
The Jones matrix associated to the ionosphere is commonly called with a T.
(a) Scenario 1 (b) Scenario 2
(c) Scenario 3 (d) Scenario 4
Figure 3.10: The four Lonsdale scenarios. Where A is the interferometer maximum baseline, V is the the beam size and S is the typical size of an ionospheric irregularity.
create a refractively distorted but coherent image, since the direction dependent effect is the same for all elements. This means that in Eq. 3.21,E term depends on (l,m) but not on (p,q) and it is a scalar again.
Scenario 4 (Fig. 3.10d) the elements have a large FoV and the array has at least some long baselines. This is the LOFAR situation and a clear example of direction dependent effect.
This scenario implies that distinct complex gain corrections may be required for each source and each receiving element. The techniques to deal with this situation are outlined in Sec. 3.3.4.
3.3 The lowest frequencies 67