{"id":4075,"date":"2025-10-28T07:09:18","date_gmt":"2025-10-28T07:09:18","guid":{"rendered":"http:\/\/www.astro.gla.ac.uk\/users\/eduard\/cesra\/?p=4075"},"modified":"2025-10-28T07:09:18","modified_gmt":"2025-10-28T07:09:18","slug":"noise-in-maps-of-the-sun-at-radio-wavelengths-by-t-s-bastian-et-al","status":"publish","type":"post","link":"https:\/\/cesra.net\/?p=4075","title":{"rendered":"Noise in Maps of the Sun at Radio Wavelengths by T. S. Bastian et al."},"content":{"rendered":"

The Sun is a powerful source of radio emissions, so much so that \u2013 unlike most celestial sources \u2013 this emission dominates the system noise of many radio telescopes. The noise resulting from such sources is referred to as \u201cself-noise\u201d. Two recent papers discuss self-noise in maps of the Sun at radio frequencies formed using the techniques of Fourier synthesis imaging. Examples of radio telescopes that exploit this technique include the LOw Frequency ARray <\/em>(LOFAR), the Jansky Very large Array <\/em>(JVLA), the Nan\u00e7ay Radiohelio- graph, <\/em>and the Expanded Owens Valley Solar Array <\/em>(EOVSA). They show that self-noise represents a fundamental limit to the dynamic range with which strong source sources may be imaged.<\/p>\n

\"\"<\/p>\n

Figure 1:<\/strong> Comparison of model sources observed by EOVSA, the 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 floor, and the green symbols trace $\\sigma_n$. Top row: the map and map rms observed 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 differences in scale for the ordinate.<\/em><\/p>\n

Paper I<\/a><\/u> develops an intuitive theory of self-noise using the limiting cases of strong point sources or strong extended sources as observed by a single dish, a two-element interferometer, and the general case of an $n$-element array of arbitrary antenna size. Workers often characterize system noise in terms of system temperature $T_{\\rm sys}$ or the source equivalent flux density $N=T_{\\rm sys}\/K$ where $K=A_e\/2k_B$; $A_e$ is the effective area of the antenna and $k_B$ is Boltzmann\u2019s constant. It is convenient to compare the source flux density $S$ with $N$, where $S$ is related to the antenna temperature $T_{ant}$ through $S=T_{\\rm ant}\/K$. When $S<<N$ the source makes a negligible contribution to system noise. This is the case for most modern radio arrays. In this case, the noise is uniform across the map. When $S>>N$, however, the source dominates the system noise and manifests itself in potentially complex ways. Paper I distinguishes between two arrays that are referred to as a \u201ccorrelation array\u201d and a \u201ctotal power array\u201d. The former is the conventional approach to Fourier synthesis imaging in the sense that auto-correlations (total power) measurements are discarded. The noise variance only includes correlations between visibility measurements. In the latter, total power measurements are retained and, when computing the noise variance, correlations between total power measurements between antennas are retained, as are correlations between total power measurements and visibilities.<\/p>\n

In the former case, the noise variance has been derived for the general case of $n$ antennas and sources of arbitrary strength by Kulkarni (1989) for snapshot integrations (i.e., for a time in which changes to the array geometry and the source are negligible). As $n$ becomes large the rms noise is given approximately by<\/p>\n

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

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 observing bandwidth and $\\tau$ is the integration time. Note that distribution of source flux is mirrored in $\\sigma_I$! The noise is therefore nonuniform across 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 antenna area $A_e$. The on-source noise is given by the full expression and the off-source noise is given by the second term, assuming the sidelobe response of the source has been approximately removed through deconvolution, a condition that cannot necessarily be met for small-$n$ arrays.<\/p>\n

In the case of a total power array, the noise is instead given by the exact expression:<\/p>\n

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

where $[\\mathcal{I}^\\circ_D]$ is the map formed from visibility data and total power. We see that in this case, the noise floor is just $\\sigma_I=N\/nM$. The self-noise on locations within the map that are free of emission can be entirely removed through deconvolution and the formal dynamic range can be very high.<\/p>\n

In practice, most large, modern Fourier synthesis radio telescopes are correlation arrays designed to address a broad program of astrophysics for which the source does not dominate the system noise ($S<<N$) and so the noise properties of maps for most cosmic sources are uniform across the map and are determined by $N$. The expense of making total power measurements on each antenna and including correlations of between total power and visibility measurements is unnecessary and unjustified in this case since it is only for cases when $S>>N$ that such measurements may be of interest.<\/p>\n

Paper II considers the implications of self-noise for a number of solar science \u201cuse cases\u201d. These include observations of the quiet (non-flaring) Sun, active regions, small transients, radio bursts, and flares. Since all extant radio arrays are correlation arrays, the first expression above is relevant. Expressing the noise in terms of brightness temperature sensitivity $\\sigma_T$ we recast the equation in terms of brightness temperature $T_b$, $T_{\\rm ant}$ and $T_{\\rm sys}$ and find that<\/p>\n

$\\sigma_T(\\theta_x,\\theta_y)\\approx \\frac{1}{M}$ $\\Bigl[T_b(\\theta_x,\\theta_y)+\\frac{T_{\\rm ant}+T_{\\rm sys}}{\\sqrt{2n_b}} \\frac{\\lambda^2}{A_e}\\frac{1}{\\Omega_{\\rm bm}}\\Bigr]\\nonumber$\u00a0$\\approx\\frac{1}{M}$ $\\Bigl[T_b(\\theta_x,\\theta_y)+\\frac{T_{\\rm ant}+T_{\\rm sys}}{(nA_e\/d^2)} \\Bigr]$<\/p>\n

where $\\Omega_{\\rm bm}\\approx (\\lambda\/d)^2$ is the angular resolution of the array, \u00a0$\\lambda^2\/A_e$ is the field of view of the instrument, and $d$ is dimension of the array (e.g., its diameter for a array with a circular footprint). One can think of $nA_e\/d^2$ as an array filling factor.<\/p>\n

There are a number of subtleties that come into play when considering various use cases. For example, solar radio bursts and flares require snapshot imaging and the on-source dynamic range is always $\\lesssim{M}$. In the case of quiet Sun imaging, 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.<\/p>\n

Conclusions<\/strong><\/p>\n

Self-noise represents a fundamental limit to the sensitivity of Fourier synthesis arrays when the source flux density $S$ is much larger than the system noise $N$. We find that it is always relevant to solar observations although the details depend on the science use case in question.<\/p>\n

Details may be found in Paper I<\/a><\/u>, which outlines theoretical considerations, and in Paper<\/a> II<\/u>, which discusses a variety of solar use cases. The corresponding arXiv reprints may be found here<\/a> and here<\/a>.<\/p>\n

References<\/strong><\/p>\n

Bastian, T. S., Chen, B., Mondal, S., & Saint-Hilaire, P., 2025a, SoPh, 300, issue 7, id. 91, doi: 10.1007\/s11207-025-02499-9<\/a><\/p>\n

Bastian, T. S., Chen, B., Mondal, S., & Saint-Hilaire, P., 2025b, SoPh, 300, issue 7, id. 90, doi: 10.1007\/s11207-025-02498-w<\/a><\/p>\n

Kulkarni, S., 1989, AJ, 98, 1112, doi: 10.1086\/115202<\/a><\/p>\n

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The Sun is a powerful source of radio emissions, so much so that \u2013 unlike most celestial sources \u2013 this emission dominates the system noise of many radio telescopes. The noise resulting from such sources is referred to as \u201cself-noise\u201d. Two recent papers discuss self-noise in maps of the Sun at radio frequencies formed using the techniques of Fourier synthesis imaging. Examples of radio telescopes that exploit this technique include […]<\/a><\/p>\n<\/div>","protected":false},"author":1,"featured_media":4076,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[66],"class_list":["post-4075","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-nuggets","tag-solar-radio-emission"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t