Description of the
method:
The software is designed to find a model-independent (regularised) flux of either photons or electrons using RHESSI count flux.
FROM COUNTS TO ELECTRONS:
The bremsstrahlung produced photon flux spectrum I(eps) [photons/sec/cm^2/keV] at the Earth is given by the integral I(eps)= 1/(4*!Pi*1AU^2)*\int_{eps}^{infinity}S(eps,e)F(e)de < /STRONG> or in matrix form
I=S*F
where F(e) = <V*n*Fe(e)>, F(e) - density and volume weighted mean electron flux spectrum , [electrons/sec/cm^2/keV], n - average number density of the surrounding plasma , [cm^-3], V - average emitting volume , [cm^3], S(eps,e) - differential in photon energy bremsstrahlung cross-section , [cm^2/keV], Fe(e) - mean electron flux spectrum [electrons/sec/cm^2/keV]
The corresponding count flux spectrum in the X-ray detectors C(eps) [counts/sec/cm^2/keV] can be written in matrix form
C = A*F, where A = DRM*S < /STRONG>
where we know C and A and want to find F . This problem is ill-conditioned and regularision (additional constraints) should be applied.
The code finds the solution for the problem as a function of lambda:
||A*F-C||^2+lambda*||L*(F-F0 )||^2=min< /STRONG>
where L - constraint matrix, F0 - thin-target fit from SPEX/OSPEX, lambda - regularisation parameter found using discrepancy principle:
||A*F(lambda)-C||=reg_tweak*||dC||
where dC is 1 sigma uncertanty on count flux, [counts/sec/cm^2/keV]
FROM COUNTS TO
PHOTONS
The observed count flux spectrum in the X-ray
detectors C
(eps) [counts/sec/cm^2/keV] can be written in matrix
form
C = DRM*
F
where DRM - dynamic response matrix of instrument. We know C and DRM and want to find a photon flux F [counts/sec/cm^2/keV].
This problem is
ill-conditioned and regularization (additional constraints) should be
applied.The code finds the solution for the problem as a
function of lambda
||DRM*F-C||^2+lambda*||L*(F-F0)||^2=min
where ||...|| - L2 norm, L- constraint matrix, F0 - a model of photon spectrum from SPEX/OSPEX, lambda - regularisation parameter found using descripancy principle:
||A*F(lambda)-C||=reg_tweak*||dC||
where dC is 1 sigma uncertainty on count flux, [counts/sec/cm^2/keV]
Last updated 11/11/2005