Effect of Unknown Inclinations in the Mass Distribution of Exoplanets

1
Effect of Unknown Inclinations in the Mass
Distribution of Exoplanets
Roman V. Baluev Saint Petersburg State University
Introduction
It is well known that routinely the true mass M
of an exoplanet cannot be determined from radial
velocity observations. The determinable parameter
is the lower bound mM sin i, where the orbital
inclination i remains unconstrained. While the
individual planetary masses cannot be determined
by radial velocities only, the exoplanetary mass
distribution may be corrected for this unknown
inclination' effect.
Let us assume that the orbital inclinations are
distributed isotropically. Remind that in this
case, the probability density functions f(M) and
g(m) may be connected by a pair of Abelian-like
integral equations
Unfortunately, it is impossible to apply the last
equality explicitly to obtain the true mass
distribution. The integro-differential operator
in this equality is unbounded. Hence, natural
statistical noise polluting the input
(observable) function g(m), leads to unbounded
random fluctuations in the output function
f(M). This means that the stated problem is
ill-posed and should be solved in some other
(more soft') way.
Method
In this work we use an approach of wavelet
analysis and recovering of probability density
functions. Such algorithm was described in the
review by Abramovich F., Bailey T.C., Sapatinas
T. (The Statistitian, 2000, V.49 P.1) for the
case of descrete wavelet transforms and was
extended to the continuous framework by Baluev
R.V. (Bull. St.Petersburg State Univ., 2008,
V.1). Let us adopt the following definition of
the Gaussian wavelet transforms of some function
p(x)
Remind that the output function W(a,b) yields a
decomposition of the input one p(x) into a set of
structures of different positions b and scales a,
but having certain constant shape. This shape is
determined by the wavelet ?j being used. For
instance, we should use the first-order Gaussian
wavelets (j1) to highlight sudden cut-offs and
jumps in the function under consideration. To
look for peaks and gaps of p(x), we should use
the second-order Gaussian wavelet.
The minimum-mass distribution g(m) differs from
the true-mass distribution f(M). So do their
wavelet transforms. However, the nature of
wavelet transforms allows an elegant way to
exclude the effect of an unknown inclination.
Namely, It is possible to perturbate the shape of
wavelets so that the respective transform of the
function g(m) will coincide with the usual
wavelet transform of f(M). Such pseudo-wavelet'
transform of g(m) is not a wavelet transform,
because the shape of perturbed wavelets depends
on their scale. Note that for large scale
parameter, these pseudo-wavelets almost coincide
with initial Gaussian wavelets.
After obtaining estimations of such
pseudo-wavelet' transform of g(m) it is possible
to clear it from the statistical noise, and to
invert the result in order to obtain the
denoised' estimation of the function f(M)
itself. Two last steps may be performed in the
way similar to that described in the papers cited
above.
Data samples
I used three samples from the Extrasolar Planets
Catalog by J.Schneider (at http//exoplanet.eu).
The first (main) sample contains 178 planets
discovered (not only confirmed) by radial
velocities around stars with masses between 0.7
Msolar and 1.35 Msolar. The second sample
contains 106 long-period planets (cold') from
the main sample having semi-major axes larger
than 0.5 AU. The third (hot') sample contains 55
planets with semi-major axes less than 0.1 AU,
discovered either by radial velocities or by
transits.
Observational selection
Of course, we have not full access to the
distribution g(m) and hence f(M) due to selection
effects. This selection is rather complicated and
is very difficult to be accounted for precisely.
The samples described above contain planets that
were discovered by different teams using
different instruments and even different
detection methods. For now, we have to restrict
ourselves to rough estimations of lower mass
limits for which the observational bias should be
negligible. For the main and cold' samples, we
can neglect this bias if m,Mgt1-2MJup. The mass
limit for the hot' sample is determined by
typical detection thresholds of photometric
surveys, m,Mgt0.5-1MJup.
Results
For all samples, the denoised estimations of
p.d.f. of exoplanetary log-mass distributions
have simple unimodal shapes without any
significant small-scale structure. Most likely,
the slopes at large masses are not perturbed, but
the slopes at low masses are due to observational
selection. Unfortunately, we have to state that
we should not expect reliable results from the
current data, except for a very few well-known
ones. For instance, the p.d.f. of log-mass
distribution decreases rapidly for Mgt2MJup, what
results in the so-called brown dwarf desert
beyond 10MJup. In addition, there is a lack of
hot' massive planets with respect to cold'
ones. The number of extrasolar planets detected
for now is too small for a deep statistical
analysis. In addition, the observational
selection hardly unveils some very interesting
exoplanetary families. However, space projects
like CoRoT and Kepler would provide in the
nearest future larger samples of exoplanets
discovered in a uniform way and covering wider
ranges of physical characteristics. These new
discoveries would allow essentially more deep
statistical analysis of their mass distributions
with the algorithm described here or similar ones.