Photometric Calibration of the LASCO-C2 Coronagraph over 14 Years (1996 – 2009)

Abstract

We present a photometric calibration of the SOHO/LASCO-C2 coronagraph based on the analysis of all stars down to magnitude V=8 that transited its field of view during the past 14 years of operation (1996 – 2009), extending the previous work of Llebaria, Lamy, and Danjard (Icarus 182, 281, 2006). The pre-processing of the images incorporates the most recent determination of the evolution of the LASCO-C2 performances. The automatic procedure then analyzes some 260 000 images to detect, locate, and measure those stars. Aperture photometry is performed using four different aperture sizes, and the zero points (ZPs) of the photometric transformations between the LASCO-C2 magnitudes for its orange filter and the standard V magnitudes are determined after introducing a correction for the color of the stars. A new statistical method (“bootstrap”) is introduced to assess the confidence intervals of the mean yearly value of the ZPs. The correction for finite aperture required to derive the calibration coefficient for the surface photometry of extended sources is based on the reconstructed image of bright saturated stars and a robust model for the growth curve. The global temporal evolution of the sensitivity of LASCO-C2 is compatible with a continuous decrease at a rate of ≈ 0.56 % per year. However, it is better described by two separate linear variations with a discontinuity at the time of the loss of SOHO. After the resumption of normal operations in 1999, the linear decrease of the sensitivity amounts to ≈ 0.35 % per year.

Appendices

Appendix A: Photometric Analysis

Let F(i,j) be the signal expressed in DN s⁻¹ at pixel (i,j). The stellar flux F _∗ is simply given by

where the process of summing the stellar signal is carried out on the vignetted images as stated above and extends over one of the above apertures ${\mathcal{D}}_{s}$, and where ${\mathcal{V}}(i,j)$ is the inverse of the in-flight vignetting function (Llebaria, Lamy, and Bout 2004) at the location of the measured star taken at the pixel of maximum signal. Note that the variation of ${\mathcal{V}}(i,j)$ is mainly radial and does not exceed 0.003 over a few pixels.

The local noise level in the circular annulus ${\mathcal{D}}_{n}$ is quantified by the variance of the signal F(i,j) defined by

where $\overline{F(i,j)}$ denotes the mean value of the signal:

In practice, $\overline{F(i,j)}\approxeq0$ since the background has been subtracted in the pre-processing.

For a given star, the adopted mean flux $\overline{F_{\star}}$ is the weighted average of all individual measurements where the weight is defined by

Consequently, the weighted mean flux for a given star is

where n _m is the number of measurements for that star. The standard deviation σ _⋆ of the mean flux $\overline {F_{\star}}$ is given by

where the factor (n _m−1) is required in the denominator to account for the fact that $\overline{F_{\star}}$ has been determined from the data and not independently (Bevington and Robinson 2003).

The instrumental (or pseudo) magnitude $\overline{m}_{\star}$ is then defined by

to which we associate its standard deviation $\sigma_{\overline{m}_{\star}}$.

The zero points (ZPs) of the transformations between the LASCO-C2 magnitudes for the different filters and the standard V magnitudes m _⋆ are defined by

and are calculated as weighted averages of the magnitude differences $(m_{k_{\star}} - \overline{m}_{\star})$ via

where the summation extends over the total number of stars n _s used to calculate ZP and where the weight is defined by

Finally the standard deviation σ _ZP of the ZP is given by

Appendix B: Bootstrap Method

The bootstrap method (Efron 1979) allows an estimation of errors without strong assumptions by determining a confidence interval associated to the mean value of a set of values, the ZPs in our case. We briefly summarize this method below.

Let us consider a statistical situation where a random sample of size n is observed from an unknown distribution Φ, such as

where X _i is distributed according to the continuous probability distribution Φ. Let x=(x ₁,…,x _n ) be the realization of the random sample X=(X ₁,…,X _n ). Given a random variable ψ=(X,Φ), such as the mean value, we wish to estimate the sampling distribution of ψ on the basis of the observed data x. The bootstrap method for a one-sample problem involves the following steps.

• i) Construct an empirical probability distribution $\hat{\Phi}$ from the sample by affecting the same probability 1/n at each point x ₁,…,x _n (each element has the same probability to be drawn), which is the nonparametric maximum likelihood estimate of the unknown distribution Φ such that

  

• ii) From $\hat{\Phi}$, a bootstrap sample is generated by drawing a random sample of size n with $x^{\star}= (x_{1}^{\star},\ldots,x_{n}^{\star})$ being the realization of the bootstrap sample $X^{\star}= (X_{1}^{\star},\ldots,X_{n}^{\star})$.

• iii) Calculate an estimate of ψ using the bootstrap sample yielding ψ ^⋆.

• iv) Repeat steps 2 and 3 in order to create B bootstrap samples. B must be large enough (B>1000) for an estimation of a confidence interval (Efron 1982).

• v) Approximate the sampling distribution of ψ=(X,Φ) by the bootstrap distribution of $\psi^{\star}= (X^{\star},\hat{\Phi})$ with the probability of 1/B at each value $\psi_{1}^{\star},\ldots,\psi _{B}^{\star}$.

An α-level confidence interval includes all the values of ψ ^⋆ between α/2 and 1−α/2 percentiles of the bootstrap sampling distribution ψ ^⋆. Thus, the 0.05 α-level confidence interval ranges between the 0.025∗B and the 0.975∗B values of the bootstrap distribution (see Figure 6 where the dashed lines fit the boundaries of the confidence interval at $\mbox{$\alpha$-level}=0.05$) and is generally asymmetric, since we do not make any assumption about the shape of the distribution.

Appendix C: Logistic Growth Model

The logistic growth model first introduced by Verhulst (1838) estimates the trend limit of growing populations. It can model the S-shaped curve of growth of some population X. The initial stage of growth is approximately exponential; as saturation begins, the growth slows, and at maturity, it stops.

For a population growth given by

where X(t) is the population at a given time t with X(t ₀) the initial population, the unimpeded growth rate is modeled by the first term R∗X.

The second term, R∗X ²/K, expresses interference between some members of the population. This antagonistic effect is called the bottleneck and is modeled by the value of the parameter K.

The solution of this model is such that: