TARJETAS DE CONTROL

To give an example of how projection effects can enter the comparison between Υdynand ΥSSP, assume that Υdynapproximates the local intrinsic stellar mass- to-light ratio, Υdyn≈Υ∗(r). Assume further that ΥSSP is a pure projection of

Υ∗(r), i.e. ΥSSP≈ΣM/ΣL, where _Σ M ΣL (r)≡ R Υ∗(r)×ν(r) dz R ν(r) dz , (5.1)

with ΣM and ΣLdenoting the surface mass and surface brightness, respectively, ν being the intrinsic luminosity profile, and the integration being performed along the line of sight. If, for example, a galaxy’s intrinsic Υ∗ is monotonically

decreasing with radiusr, then the projected mass-to-light ratio ΣM/ΣL(R) (at projected radius R) will be smaller than the intrinsic Υ∗(R). This, because

along the line of sight r > R and, thus, Υ∗(r) < Υ∗(R) in the integral of

equation (5.1).

To quantify this effect, the projection quadratures in equation (5.1) have to be solved for different Υ∗(r). This can be done conveniently by means of orbit

libraries as calculated for each galaxy. To this end, recall that the integrated luminosity dLj _{and surface brightness dSB}j _{in binj of an orbit model read}

dLj =X i widLj_i (5.2) and dSBj =X i widSBj_i (5.3)

(cf. equation 2.5; dSBj_i is the total projected light of orbit i in bin j). The orbital weightwi equals the total amount of light carried by the corresponding orbit. Given an arbitrary Υ∗(r), a mass weightµican be assigned to orbitivia

µi≡wi×Υ∗(hrorbii), (5.4)

where hrorbii is the mean orbital radius defined in App. C. Analogously to

equations (5.2, 5.3) the intrinsic and projected mass in bin j can be expressed via

dMj =X

i

5.3 Crosscheck with stellar populations 119

and

dSMj=X

i

dSBj_iµi. (5.6)

Equations (5.2,5.3) and (5.5, 5.6), respectively, determine the projected mass- to-light ratio (in bin j)

ΣM ΣL j ≡dSM j dSBj (5.7)

the local mass-to-light ratio

ρ ν j ≡ dM j dLj (5.8)

and the cumulative mass-to-light ratio

M L j ≡Σk<jdM k Σk<jdLk . (5.9)

The sums on the right hand side of equation (5.9) are intended to comprise ev- ery binkwith radius smaller than that of the actual binj. Note, that the local mass-to-light ratioρ/νof the final orbit superposition will not necessarily equal the original Υ∗(r) of equation (5.4) exactly. Apart from noise due to finite orbit

sampling and finite bin sizes the main reason is that radially extended orbits spread the Υ∗ of their mean radius to larger and smaller radii. In this sense,

an orbit library calculated in a (not necessarily self-consistent) gravitational po- tential that allows for a superposition with much emphasis on radially confined orbits will yieldρ/ν closer to Υ∗(r) than another potential that requires a sub-

stantial fraction of radially floating orbits. For the purpose here, however, the exact shape of ρ/ν doesn’t matter but only the consistency betweenρ/ν and ΣM/ΣL. The aim is to cover the profile shapes found in Figs.5.2and 5.3with a handful of representative profiles only whose global qualities such as being monotonic or having a maximum/minimum need to be specified. In any case, application of equation (5.4) has the advantage of ensuring that the finalρ/νis supported by an orbit distribution and, thus, is stationary.

Fig. 5.6 shows exemplary for GMP5975 projected, local and cumulative mass-to-light ratio profiles for three different Υ∗(r) with piecewise constant

logarithmic gradient (i) left column:

d log Υ

d logr ≡ ±0.23 (5.10)

(ii) middle column:

d log Υ d logr =±

0.15 : r60.75reff

−0.7 : r >0.75reff (5.11) (iii) right column:

d log Υ d logr =±

0 : r60.75reff

(Υ∗ andrare scaled to solar units and the half-light radius reff, respectively). The projections are performed in the best-fit mass distribution of GMP5975. In the lower panel of the figure the fractional difference

D ≡diff ≡ΣM/ΣL_M(−_r)/M(_L(r_r)₎/L(r) (5.13) between ΣM/ΣL and M(r)/L(r) is plotted. As will become clear below it pro- vides conservative limits on the offset between Υ∗ and ΣM/ΣL.

As expected, for a monotonic Υ∗the localρ/νis bracketed between ΣM/ΣL

(steeper than Υ∗) and M(r)/L(r) (shallower than Υ∗). In case(i) Dis below

20 percent inside 0.2reff but increases to about 35 per cent at large radii. A monotonic Υ∗ with a break like in case (iii) bounds D to about 10 percent

inside 0.2reff but results in a steeper increase with radius. Finally, in case (ii) the sign change in the slope of Υ∗ is reflected in a similar sign change ofD. In

the particular case shown in the middle column of Fig. 5.6 D reaches 40 per cent in the outer parts, but remains low near the centre (positive and negative variations along the line of sight cancel out).

The fractional difference D will not only depend on Υ∗ but also on the

specific light profile of each galaxy. Ellipticals, to first order, follow similar light profiles, allowing to apply the above results to the whole sample for an order of magnitude estimate. Projection effects also depend on inclination. In the limit

i→0◦_{vertical gradients affect the projection as radial gradients do in the edge-}

on case. Vertical gradients can be generally as important as radial gradients, but are unlikely to lead to qualitatively different results. Thus, the edge-on case can be regarded as being representative to cover all possible projection effects.

The profile shapes and slope magnitudes of Fig. 5.6 roughly comprise the cases in Figs. 5.2 and 5.3, respectively. Furthermore, inspection of the blue, cyan and red curves reveals that the deviations between ρ/ν and ΣM/ΣL are commonly smaller than those between M(r)/L(r) and ΣM/ΣL (this holds at least out to reff). Thus, the above defined Dlimits for the Coma galaxies the possible bias between Υdynand ΥSSP related to pure projection effects.