We investigate the ability of three-integral , axisymmetric , orbit-based modeling algorithms to recover the parameters defining the gravitational potential ( mass-to-light ratio \Upsilon and black hole mass M _ { \bullet } ) in spheroidal stellar systems using stellar kinematical data .
We show that the potential estimation problem is generically under-determined when applied to long-slit kinematical data of the kind used for most black hole mass determinations to date .
A range of parameters ( \Upsilon,M _ { \bullet } ) can provide equally good fits to the data , making it impossible to assign best-fit values .
The indeterminacy arises from the large variety of orbital solutions that are consistent with a given mass model .
We demonstrate the indeterminacy using a variety of data sets derived from realistic models as well as published observations of the galaxy M32 .
The indeterminacy becomes apparent only when a sufficiently large number of distinct orbits are supplied to the modeling algorithm ; if too few orbits are used , spurious minima appear in the \chi ^ { 2 } ( \Upsilon,M _ { \bullet } ) contours , and these minima do not necessarily coincide with the parameters defining the gravitational potential .

We show that the range of degeneracy in M _ { \bullet } depends on the degree to which the data resolve the radius of influence r _ { h } of the black hole .
For { FWHM } / 2 r _ { h } \gtrsim 0.5 , where FWHM refers to the instrumental resolution , we find that only very weak constraints can be placed on M _ { \bullet } .
In the case of M32 , our reanalysis demonstrates that when a large orbit library is used , data published prior to 2000 ( { FWHM } / 2 r _ { h } \approx 0.25 ) are equally consistent with black hole masses in the range 1.5 \times 10 ^ { 6 } { { \cal M } _ { \odot } } < M _ { \bullet } < 5 \times 10 ^ { 6 } { { \cal M } _ { \odot } } , with no preferred value in that range .
Exactly the same data can reproduce previous published results with smaller orbit libraries .
While the HST/STIS data for this galaxy ( { FWHM } / 2 r _ { h } \approx 0.06 ) may overcome the degeneracy in M _ { \bullet } , HST data for most galaxies do not resolve the black hole ’ s sphere of influence and in these galaxies the degree of degeneracy allowed by the data may be greater than previously believed .

We investigate the effect of regularization , or smoothness constraints , on the degree of degeneracy of the solutions .
Enforcing smoothness reduces the range of acceptable models , but we find no indication that the true potential can be recovered simply by enforcing smoothing .
For a given smoothing level , all solutions in the minimum- \chi ^ { 2 } valley exhibit similar levels of noise ; as the smoothing is increased , there is a systematic shift in the midpoint of the \chi ^ { 2 } valley , until at a high level of smoothing the solution is biased with respect to the true solution .
These experiments suggest both that the indeterminacy is real – i.e. , that it is not an artifact associated with non-smooth solutions – and that there is no obvious way to choose the smoothing parameter to ensure that the correct solution is selected .