Noise maps of the Sun at radio wavelengths by TS Bastian et al. – Community of European Solar Radio Astronomers

The Sun is a powerful source of radio emissions, to the point that, unlike most celestial sources, this emission dominates the system noise of many radio telescopes. The noise resulting from these sources is called “own noise”. Two recent papers analyze self-noise in maps of the Sun at radio frequencies formed using Fourier synthesis imaging techniques. Examples of radio telescopes that exploit this technique include the Low frequency array (LOFAR), the Very large Jansky matrix (JVLA), the Nançay radioheliograph, and the Owens Valley Expanded Solar Panel (EOVSA). They show that self-noise represents a fundamental limit to the dynamic range with which powerful sources can be imaged.

Figure 1: Comparison of model sources observed by EOVSA, JVLA and the proposed ngVLA at a nominal frequency of 6 GHz. In each panel, the dashed blue line represents $\mathcal{I}_D$, the solid blue line is $|\mathcal{I}_D+S/\sqrt{2n_b}|$, the dashed red line represents the noise level, and the green symbols plot $\sigma_n$. Top row: the map and the observed rms map for a point source with $S = 1$; Second row: the same for a point source with a flux density $S_{\rm pt} = 0.2$ and a total flux $S_{\rm pt} +S_{\rm bg} = 1$; Third row: The same for a Gaussian source with $\theta_G = 30″$ and a total flux $S = 1$. In all cases $N=0$. Note the scale differences for the ordinates.

Paper I develops an intuitive theory of self-noise using the limiting cases of strong point sources or strong extended sources observed by a single dish, a two-element interferometer, and the general case of an $n$ element array of arbitrary antenna size. Workers typically characterize system noise in terms of the system temperature $T_{\rm sys}$ or the equivalent flux density of the source $N=T_{\rm sys}/K$ where $K=A_e/2k_B$; $A_e$ is the effective area of ​​the antenna and $k_B$ is the Boltzmann constant. It is convenient to compare the flux density of the source $S$ with $N$, where $S$ is related to the antenna temperature $T_{ant}$ through $S=T_{\rm ant}/K$. When $S>N$, however, the source dominates the system noise and manifests itself in potentially complex ways. Article I distinguishes between two sets referred to as the “correlation set” and the “total power set.” The first is the conventional approach to Fourier synthesis imaging in that autocorrelation (total power) measurements are discarded. Noise variation only includes correlations between visibility measurements. In the latter, total power measurements are retained and, when calculating the noise variation, correlations between total power measurements between antennas are retained, as are correlations between total power measurements and visibilities.

In the first case, Kulkarni (1989) derived the noise variance for the general case of $n$ antennas and sources of arbitrary intensity for instantaneous integrations (i.e., for a time when changes in array and source geometry are negligible). As $n$ becomes large, the rms noise is given approximately by

$\sigma_I(\theta_x,\theta_y)\approx \frac{1}{M} \Bigl[ \mathcal{I}_D(\theta_x,\theta_y)+\frac{S+N}{\sqrt{2n_b}} \Bigr].$

where $n_b=n(n-1)/2$ is the number of antenna baselines in the array and $M=\sqrt{\Delta\nu\tau}$; $\Delta\nu$ is the observation bandwidth and $\tau$ is the integration time. Note that the source flow distribution is reflected in $\sigma_I$. Therefore, the noise is not uniform throughout the map. It is also important to note that a uniform noise floor is always present: $(S+N}/{\sqrt{2n_b}\approx S/nM$. Finally, note that the noise is independent of the antenna area $A_e$. The noise at the source is given by the full expression and the noise outside the source is given by the second term, assuming that the sidelobe response of the source has been approximately removed by deconvolution, a condition that is not can necessarily be satisfied for small $n$ matrices.

In the case of a total power matrix, the noise is given by the exact expression:

$\sigma_I (\theta_x,\theta_y)=\frac{1}{M} \Bigl[\mathcal{I}^\circ_D(\theta_x,\theta_y)+\frac{N} {n}\Bigr].$

where $[\mathcal{I}^\circ_D]$ is the map formed from visibility and total power data. We see that in this case, the background noise is simply $\sigma_I=N/nM$. Self-noise at locations within the map that are emission-free can be completely removed by deconvolution and the formal dynamic range can be very high.

In practice, most modern large Fourier synthesis radio telescopes are correlation arrays designed to address a broad astrophysics program for which the source does not dominate the system noise ($S>N$ that such measurements may be of interest.

Paper II considers the implications of self-noise for a series of “use cases” in solar science. These include observations of the quiet Sun (without flares), active regions, small transients, radio bursts and flares. Since all existing radius sets are correlation sets, the first expression above is relevant. Expressing the noise in terms of sensitivity to brightness temperature $\sigma_T$ we reformulate the equation in terms of brightness temperature $T_b$, $T_{\rm ant}$ and $T_{\rm sys}$ and find that

$\sigma_T(\theta_x,\theta_y)\approx \frac{1}{M}$ $\Bigl[T_b(\theta_x,\theta_y)+\frac{T_{\rm ant}+T_{\rm sys}}{\sqrt{2n_b}} \frac{\lambda^2}{A_e}\frac{1}{\Omega_{\rm bm}}\Bigr]\nonumber$ $\approx\frac{1}{M}$ $\Bigl[T_b(\theta_x,\theta_y)+\frac{T_{\rm ant}+T_{\rm sys}}{(nA_e/d^2)} \Bigr]$

where $\Omega_{\rm bm}\approx (\lambda/d)^2$ is the angular resolution of the array, $\lambda^2/A_e$ is the field of view of the instrument, and $d$ is the dimension of the array (e.g., its diameter for an array with a circular footprint). $nA_e/d^2$ can be thought of as a matrix filling factor.

There are a number of subtleties that come into play when considering various use cases. For example, solar radio bursts and flares require instantaneous imaging and the dynamic range at the source is always $\lesssim{M}$. In the case of quiet images of the Sun, the noise is well approximated by a uniform noise floor. In this case, it may be possible to exploit Earth rotation aperture synthesis and/or multi-frequency synthesis.

Conclusions

Self-noise represents a fundamental limit to the sensitivity of Fourier synthesis matrices when the source flux density $S$ is much larger than the system noise $N$. We found that it is always relevant to solar observations, although the details depend on the scientific use case in question.

Details can be found at Paper Iwhich describes theoretical considerations, and in Paper IIwhich looks at a variety of solar use cases. Corresponding arXiv reprints can be found here and here.

References

Bastian, TS, Chen, B., Mondal, S. and Saint-Hilaire, P., 2025a, SoPh, 300, issue 7, id. 91, doi: 10.1007/s11207-025-02499-9

Bastian, TS, Chen, B., Mondal, S. and Saint-Hilaire, P., 2025b, SoPh, 300, issue 7, id. 90, doi: 10.1007/s11207-025-02498-w

Kulkarni, S., 1989, AJ, 98, 1112, doi: 10.1086/115202