# RMCSat: An F10.7 Solar Flux Index CubeSat Mission
## 2 F10.7 Index  
The measurement of solar flux is based on the fundamentals of blackbody radiation. According to the Stefan-Boltzmann law, a perfect blackbody will emit energy based on its surface area \(A\) and its absolute temperature \(T\) in Kelvin. The total isotropic power \(P\) emitted across all frequencies is given by,  
\[\frac{P}{A}=\sigma T^{4}, \tag{1}\]  
where \(\sigma=5.67\times 10^{-8}\frac{W}{m^{2}K^{4}}\) is the Stefan-Boltzmann constant. The spectral radiance of an object of a given temperature does vary with frequency. When observed at a particular frequency, the spectral radiance \(B_{v}\) of emissions from an object follows Planck's Law,  
\[B_{v}=\frac{2 hv^{3}}{c^{2}}\frac{1}{e^{\frac{hv}{k_{B}T}}-1}, \tag{2}\]  
where \(v\) is the wavelength of emission, \(h=6.63\times 10^{-34}\) Js, is the Planck constant and \(k_{B}=1.38\times 10^{-23}\frac{m^{2}kg}{s^{2}K}\) is Boltzmann constant. In the case of microwave frequencies at solar temperatures, where \(hv~{}\ll~{}k_{B}T\), the Rayleigh-Jeans approximation can be used to simplify Equation (1) for solar radiation analysis where,  
\[B_{v}=\frac{2k_{B}Tv^{2}}{c^{2}}=\frac{2k_{B}T}{\lambda^{2}}. \tag{3}\]Spectral radiance is a characteristic of the emitting source, the Sun in this instance, while intensity is what is observed by a receiver, the antenna. The specific intensity \(I_{v}\) is given as  
\[I_{v}=\frac{2k_{B}T_{b}v^{2}}{c^{2}}=\frac{2k_{B}T_{b}}{\lambda^{2}}. \tag{4}\]  
If the specific intensity of the radiation source is integrated over the total solid angle subtended by the emitting source \(\Omega_{s}\), as measured at the detector, the result is the spectral flux density \(S_{v}\),  
\[S_{v}=\int I_{v}cos\theta d\Omega. \tag{5}\]  
By using the small-angle approximation for the cosmic radio sources,  
\[S_{v}=\int I_{v}d\Omega=I_{v}\;\Omega_{s}. \tag{6}\]  
Given Equation (4), the spectral flux density becomes,  
\[S_{v}=\frac{2k_{B}T_{b}\;\Omega_{s}}{\lambda^{2}}, \tag{7}\]  
where \(S_{v}\) is the power per unit area per unit bandwidth due to the source measured at wavelength \(\lambda\). This is the measurement of solar flux density commonly used for solar activity analysis and is the basis for F10.7 measurements which focus on an average wavelength of 10.7 cm. The F10.7 index is the total power flux density, \(S_{v}\), from all sources on the solar disk at 10.7 cm within a bandwidth of 100 MHz centered on 2800 MHz, averaged over an hour [1]. It is comprised of three components: a constant one from quiet Sun background emissions; a slowly varying component (S-component) that originates in active regions and; and a highly variable one resulting from radio burst emissions [1; 10]. The recorded flux density at this wavelength varies daily. It is expressed in solar flux units (sfu) or \(10^{-22}\) W/mHz and can vary from approximately 64 fsu to greater than 300 sfu over the solar cycle [1]. Flux measurements taken from the ground-based DRAO are obtained using two 1.8 m diameter paraboloid radio telescopes that run in parallel. They are referred to as Flux Monitor 1 and Flux Monitor 2; the latter acts as the primary instrument while the former is a backup [1]. During the northern hemisphere summer months, measurements are taken each day at 1700, 2000, and 2300 UTC, and in winter at 1800, 2000, 2200 UTC due to the reduced daylight duration. Readings are taken both on-Sun and off-Sun, to obtain accurate background cold-sky values. The solar data are then corrected for atmospheric absorption, background sky temperature, and antenna gain [17]. Precise calibration standards are maintained through annual calibration and validation of both flux monitors against an external reference source, a large pyramidal horn antenna maintained onsite at the DRAO.