Appendix A: Calculation of the envelope of the Spaenhauer diagram

We consider the population of sosie galaxies for a calibrator of absolute magnitude M₀. The distribution function of its population Nis assumed to be Gaussian $\mathcal{G}(M_{\rm o},\sigma)$; The probability for a galaxy of this population, to have an absolute magnitude M_i, with M_i<M' is given by:

$$P(M_{i}<M')=\frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^{M'}\exp{\left(-\frac{(x-M_{0})^2}{2{\sigma}^2}\right)} {\rm d}x.$$

Figure A.1: Several Gaussian functions with the same standard deviation $\sigma$and different populations N. The actual range n x around the mean depends on the population.

The difficulty is illustrated in Fig. A.1. With the same standard deviation, the number of points beyond a given limit $r\sigma$ increases with the population N. Our goal is to take into account this fact to calculate the equation of the curve enveloping the points of the Spaenhauer diagram.

Let us write the probability of having a galaxy with $M_{i}>M_{0}+r\sigma$. We will consider only this case but obviously, there is a similar case on the other side of the Gaussian distribution.

$$P(M_{i}>M_{0}+r\sigma)=\frac{1}{\sigma\sqrt{2\pi}}\int_{M_{0}+r\sigma}^{\infty}\exp{\left(-\frac{(x-M_{0})^2}{2{\sigma}^2}\right)}{\rm d}x.$$

It is useful to take the centered variable u=x-M₀:

 

$$P(M_{i}-M_{0}> r\sigma)=\frac{1}{\sigma\sqrt{2\pi}}\int_{r\sigma}^{\infty}\exp{\left(-\frac{u^2}{2{\sigma}^2}\right)}{\rm d}u$$ (A.1)

For a population of N observed galaxies, the probability for one of them to be at $r\sigma$ from M₀ is:

 

$$P\left(\bigcup_{i=1}^N{[M_{i}-M_{0} > r\sigma]}\right) N.P(M_{i}-M_{0} > r\sigma)$$ = (A.2)

because events are independent and have the same probability.

The criterion defining the upper limit of the points in the Spaenhauer diagram is simply:

$$P\left(\bigcup_{i=1}^N{[M_{i}-M_{0} > r\sigma]}\right) < \epsilon$$

where $\epsilon$ is an arbitrary probability level.

Then, using Eqs. (A.1) and (A.2), we obtain:

$$\frac{N}{\sigma\sqrt{2\pi}}\int_{r\sigma}^{\infty}\exp{\left(-\frac{u^2}{2{\sigma}^2}\right)}{\rm d}u < \epsilon.$$ (A.3)

With $v=u / \sigma\sqrt{2}$, the envelope equation becomes:

$$\frac{N}{\sqrt{\pi}}\int_{a}^{\infty}\exp{(-v^2)}{\rm d}v = \epsilon$$

where $a={r}/{\sqrt{2}}$. After integration, we obtain:

$$\frac{N}{2}\left[1-erf\left(\frac{r}{\sqrt{2}}\right)\right] = \epsilon$$ (A.4)

i.e.,

$$\log~N = -\log ({\rm erfc}(\frac{r}{\sqrt{2}}) + \log (2\epsilon).$$ (A.5)

Figure A.2: Representation of Eq. (A.5) for an arbitrary probability level $\epsilon = 0.05$. The curve (solid line) is the true equation. The dashed line represents our linear representation for a population between about 10 and 3000 galaxies.

Figure A.2 shows the shape of the curve and the linear representation we adopted over the population range 10 to 3000. With $\epsilon = 0.05$, this linear representation is:

$$\log~N=1.55r-3.08.$$

In fact, with a normalization procedure, we need only the slope which does not depend on $\epsilon$. The number of galaxies observed up to a distance d is:

$$N= \frac{4}{3}\pi \rho d^{\alpha}$$

where $\rho$ is the space density of the considered galaxies and $\alpha$ is the fractal dimension describing the distribution of galaxies ($\alpha= 3$ in the case of homogeneous distribution). Using the Hubble law (d=V/H), we obtain:

$$\log~N ={\alpha }\log V + {\rm constant}(H,\rho).$$

Hence, the points of the envelop must fulfil simultaneously:

$$\vert M-M_{0}\vert = r\sigma$$

$$\log~N=1.55r+b(\epsilon)$$

$$\log~N={\alpha} \log v+b'(H,\rho)$$

where b depends on $\epsilon$ only and b' depends on $\rho$, H.

The equation of the envelop obtained is then:

$$\vert M-M_{0}\vert = \frac{\alpha\sigma}{1.55} \log V + k.$$ (A.6)

The constant k depends on the values of $\rho$, H. We can find k (normalization) if we know a point of the curve. This is done (see the main text) by imposing that the intersection of the completeness curve with the envelope has an abscissa V_0.05.