Solar flares are the most vigorous explosive phenomena in our solar system. They release up to $\sim10^{33}$ erg of magnetic energy in the Sun's corona in times that range from minutes to hours. Some 10 to 50\% of the flare energy goes into electron acceleration. Among other processes, when these electrons interact with the ambient plasma, they produce bremsstrahlung radiation in hard X-rays (HXRs). Analyses of flare HXRs are critical for understanding energy release dynamics, acceleration mechanisms, and their connection with other phenomena in the corona. One of these phenomena is coronal heating, an open problem in heliophysics. This problem seeks to clarify why the Sun's coronal temperature is up to three orders of magnitude higher than that at the Sun's surface.
Coronal temperatures demand a mean energy input between $\sim10^5$ and $2\times10^7$ erg cm$^{-2}$ s$^{-1}$. Multiple observations have proven that medium and large-size flares together do not contribute enough energy to account for these input power requirements. Instead, a popular idea proposes that the solar atmosphere is filled with small impulsive heating events releasing magnetic energy in the corona, called nanoflares. If nanoflares follow the same physics as their larger counterparts, they should emit hard X-rays (HXRs) but with substantially fainter intensity. A copious and continuous presence of nanoflares would result in sustained HXR emission. These nanoflares could deliver sufficient energy into the Sun's corona, to account for its high temperatures. To date, there has not been any direct detection of such persistent HXRs emitted from the quiescent Sun. However, $\sim12$ days of solar off-pointing observations of the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) during periods of quiescent activity led to HXR upper limits. In the 6-12 keV energy range, e.g., this upper limit is $9.5\times 10^{-4}$ photons s$^{-1}$ cm$^{-2}$ keV$^{-1}$.
Observing faint HXR emission is challenging because it demands instruments with high sensitivity and dynamic range. RHESSI has insufficient sensitivity to detect such faint sources, especially in the form of a broad, diffuse signal rather than the bright, compact signals for which RHESSI was designed. The Focusing Optics X-ray Solar Imager (FOXSI) sounding rocket experiment excels in these two attributes. FOXSI has a sensitivity of $\sim$ 0.0032 photons cm$^{-2}$ s$^{-1}$ keV$^{-1}$ ($\sim$50 times that of RHESSI) at 8 keV and a dynamic range of $\sim$100 for sources $>30$ arcsec apart. FOXSI achieves such a superior performance by pairing nested grazing-incidence Wolter-I mirrors with low-noise semiconductor detectors optimized for high energies. FOXSI's direct focusing capabilities allow quiet regions of the corona to be isolated to look for the presence of HXR sources.
This thesis constrains the quiet Sun emission in the 5-10 keV energy range using FOXSI observations from the second and third rocket flights (FOXSI-2 and -3). To fully characterize FOXSI's sensitivity, this thesis presents a thorough optics calibration and a ray-tracing simulation to assess ghost ray backgrounds generated by sources outside of the telescope field of view. This work demonstrates a Bayesian approach to provide upper thresholds of quiet Sun HXR emissions and probability distributions for the expected flux of a quiet-Sun HXR source when it is assumed to exist. For FOXSI-2 and -3, such upper limits are $4.5\times 10^{-2}$ photons s$^{-1}$ cm$^{-2}$ keV$^{-1}$ and $6.0\times10^{-4}$ photons s$^{-1}$ cm$^{-2}$ keV$^{-1}$, respectively (both in the 5-10 keV energy range). These two limits are similar to that of RHESSI in the 6-12 keV energy band ($9.5\times 10^{-4}$ photons s$^{-1}$ cm$^{-2}$ keV$^{-1}$) but with an important difference: it took $\sim$1/2600 less integration time for FOXSI to get enough statistics to yield these equivalent limits. The FOXSI-2 limit presented in this doctoral work is the first-ever quiet Sun upper threshold in HXR estimated from observations performed during a period of high solar activity. This dissertation's quiet Sun HXR analyses during a solar cycle minimum are the first scientific results that use the $\sim6.5$ minutes of the FOXSI-3 rocket observations. A possible future spacecraft using FOXSI's concept would allow enough observation time to constrain the current HXR quiet Sun limits further or perhaps even make direct detections. This last objective would demand observations of a few hours at the very least and (ideally simultaneous) onboard measurements of the backgrounds. Any upper quiet Sun HXR limit constrains the parameter space (e.g., the index and cutoff energy for thick target power-law models) that nanoflare electron energy distribution can have. The limits found in this doctoral work suggest very steep spectra, i.e., $>5$ power-law indexes when we assume nanoflare accelerated electrons follow a thick target model (in agreement with earlier RHESSI-based results).
