XMM-Newton Science Analysis System


eposcorr (eposcorr-3.13) [xmmsas_20230412_1735-21.0.0]

Statistical method

The statistic for optimizing the match between optical and X-ray sources is:

  $$L = \sum_{i=1}^{n_x} \sum_{j=1}^{n_o} \exp( -\frac{1}{2} (\frac{r_{ij}}{\sigma_{ij}} )^2 ),$$ (1)

with $r_{ij}$ the distance between an X-ray (i) and an optical source (j), $\sigma_{ij}$ the associated error and $n_x$ resp. $n_o$ the number of X-ray sources and optical sources in the list. In the eposcorr task only those optical sources are considered which are within $5\sigma$ of an X-ray source (for a given position offset).

In the following we will assume that the errors in the RA and DEC are equal and uncorrelated and follow a Gaussian distribution, we can then write $\sigma_x = \sigma_y = \sigma$. For the expectation value of $L$ for a given X-ray source we get:

  $$L = \frac{1}{2\pi \sigma^2} \int dx \int dy \exp( -\frac{1}{2} \... ...2\pi \sigma^2} \int dr\ 2\pi r\ \exp( - \frac{r^2}{\sigma^2} ) = \frac{1}{2}.$$ (2)

The associated variance in $L$ is:

  $$<L^2>-<L>^2 = \int dr\ 2\pi r\ \exp( - \frac{1}{2}\frac{r^2}{\sig... ... \Bigr)^2-\Bigl(\frac{1}{2}\Bigr)^2 = \frac{1}{3} - \frac{1}{4} = \frac{1}{12}$$ (3)

Of course, in practice there will be chance coincidences. For chance coincidences the chance that an optical counter part will be within a distance $r$ from the source is $\pi r^2/ \pi(5\sigma)^2$ (i.e. within an error circle of $5\sigma$). This gives for the expected value $L$ for a chance coincidence:

  $$L = \int_0^{5\sigma} dr \frac{2\pi r}{\pi 25 \sigma^2} \exp( - \f... ...^2} ) = \frac{2}{25} \Bigl(1 - \exp(-\frac{25}{2})\Bigr) \simeq \frac{2}{25}.$$ (4)

And for the variance in $L$:

  $$<L^2>-<L>^2 = \int_0^{5\sigma} dr \frac{2\pi r}{\pi 25 \sigma^2} ... ...}\Bigr)^2 \simeq \frac{1}{25} - \Bigl(\frac{2}{25}\Bigr)^2 = \frac{21}{25^2} .$$ (5)

The number of chance coincidences can be estimated using poison statistics with a poison parameter of $\mu = 25 \pi \sigma^2 \lambda$, with $\lambda$ the average number of optical sources per unit area. The expected number of sources is thus $\sum_{i=1}^{n_x} \mu_i$ (where the subscript $i$ denotes values for each X-ray source, thus allowing for fluctuations in the number of optical sources per area). The expected value for $L$ is:

  $$L = \sum_{i=1}^{n_x} \mu_i \int_{0}^{5\sigma_i} dr \ \frac{2\pi ... ...( - \frac{1}{2} \frac{r^2}{\sigma^2} ) = \frac{2}{25} \sum_{i=1}^{n_x} \mu_i .$$ (6)

How many counterparts do we need to discriminate between chance coincidences or real counter parts? This question is not easy to answer, as eposcorr optimizes $L$ and also for the number of counter parts. This means that Poissonian statistics may not be valid. To get at least an approximate answer, we equate:

  $$(L_{exp} - 2\sigma_L)_{gaussian} = (L_{exp} + 2\sigma_L)_{poissonian},$$ (7)

or,

  $$\frac{N}{2} - 2\sqrt{\frac{N}{12}} = \frac{2N}{25} + 2\sqrt{\frac{2N}{25^2}}.$$ (8)

The solution of this equation is $N = 5.1$. I therefore propose to use this number plus the number of degrees of freedom as the minimum threshold for accepting a result of eposcorr. This means that when offsets and in RA and DEC are corrected for the minimum number of optical counter parts should be 7, including a rotational correction this will be 8. This number will be contained in the keyword NMATCHES.