THE POSTERIOR DISTRIBUTION OF sin(i) VALUES FOR EXOPLANETS WITH M_T sin(i) DETERMINED FROM RADIAL VELOCITY DATA

As is well known the observational study of exoplanets began with, and in large part has been based on, radial velocity (RV) data which allow a measurement of the planet's orbital parameters plus a value of M_T sin(i), where M_T is its true mass and sin(i) is the angle between the direction normal to the planet's orbital plane and the observer's sight line (see Marcy et al. 2003, 2005; Cumming et al. 2008; Johnson 2009). Indeed, the very classification of an unseen stellar companion as an exoplanet is normally made based on the value of M_T sin(i), hereinafter designated as M₀, the observed or indicative mass.

It would, of course, be preferable to determine M_T itself and avoid the degeneracy with the largely uninteresting random variable sin(i), and our understanding of exoplanet systems has been greatly advanced by the relatively few cases in which the observations of transit events allows the two parameters to be measured separately (see Charbonneau et al. 2007 and references therein; Winn 2010 and references therein).

Nevertheless, the M_T sin(i) degeneracy does not seem too serious because it appears to be so simple and well understood. In particular, it seems extremely safe to assume that i is randomly and isotropically distributed or, in other words, that the orientation of the orbital plane of an exoplanet in space is independent of the direction from which we observe it. However, this isotropic distribution only describes the prior distribution of i, not its posterior one, i.e., not the relevant distribution after it is conditioned by the measurement of an M_T sin(i) value.

In order to specify this isotropic i prior distribution of sin(i), we consider a longitudinal strip of a sphere between θ = i and θ = i + dθ in polar coordinates. The strip extends around 2π in ϕ (the azimuthal angle), and the surface area of the strip is just 2πr² sin(i)dθ. The probability of a randomly oriented vector piercing that area is then just its fractional area of the hemispherical surface of the sphere

$$\begin{equation} f_i = 2\pi r^2 \sin (i) d\theta / 2 \pi r^2. \end{equation} \tag{ 1 }$$

The pdf of the inclination angle (assuming random orientation) is thus sin(i). In other words, the probability distribution function of the inclination angle of the exoplanet falls into this range i to i + dθ is just sin(i). (Note that we need only consider a single hemisphere since the distribution of i will be the same in both.)

In order to determine the prior probability of sin(i), simply consider

$$\begin{equation} f_i di = f_j dj, \end{equation} \tag{ 2 }$$

where j = sin(i). After some algebraic manipulation, it is easy to show that f_j (the pdf of sin(i) falling into a range of sin(i) − d sin(i) and sin(i) + d sin(i)) is $j\sqrt{1-j^2} = \tan (i)$.

Up to this point, the analysis is straightforward. However, complications arise at the next step, the derivation of the posterior distribution of sin(i), because it depends on the prior or true distribution of M_T. As we do not yet know the M_T distribution with high precision in any mass range (see Jorissen et al. 2001; Zucker & Mazeh 2001; Marcy et al. 2005; Udry & Santos 2007; Brown 2011) and have little or no empirical information constraining it in others, this consideration is an important one not only in principle but perhaps also be in practice. The present paper is primarily intended to investigate this issue, the posterior distribution of sin(i) given an observation of M₀, in some detail.

Before presenting a Bayesian analysis in the next sections, it may be helpful to note that the issue resembles familiar complications in interpreting photometric data that are conventionally called Malmquist-type biases (see Malmquist 1920; Eddington 1913; Hogg & Turner 1998) in some respects. Namely, even if the measurement errors are symmetric and unbiased (and, in the simple cases most often analyzed, also Gaussianly distributed—but that is not essential), the true brightness of an astronomical object is normally more likely to be fainter than its measured brightness than it is to be brighter. The well-known reason is that there are usually a larger number of fainter objects than brighter ones on which the (symmetrical) measurement errors may act to produce the observed brightness.

However, the considerations for sin(i) which we investigate in this paper are not related to measurement errors. It would be unchanged even if all of the observations in question were perfect and ideal. Neither is it a selection bias on sin(i) of the sort that was briefly considered as an explanation of exoplanet RV discoveries in their earliest days (see Black 1997; Gray 1997). Rather, we are considering the unavoidable consequences of the combination of a physical variable, M_T with an unobservable stochastic one, sin(i), when conditioned by a measurement of their product. This is, of course, a classical issue in Bayesian statistics.

Section 2 defines the basic question addressed by this paper and gives a very simple illustrative example of why it can be an important issue. Section 3 presents a Bayesian derivation of the equations needed to answer the question for any given distribution of masses for a population of exoplanets, and Section 4 presents the results of the analysis obtained by assuming various "toy models" for the true exoplanet distribution of masses. We then discuss observational selection effects briefly in Section 5 and conclude in Section 6 with a discussion of its practical implications for RV studies of exoplanets.

The question we wish to analyze can be formulated in two equivalent but slightly different forms, one describing the M_T distribution and one the sin(i) distribution.

• 1.  
  What is the probability that M_T is less than X, given that RV data yield M₀ (=M_T sin(i))? The answer may be written as P(M_T < X|M₀) and depends on P(M_T), the intrinsic distribution of exoplanet masses.
• 2.  
  What is the probability that sin(i) is less than Z, given that RV data yield M₀ (=M_T sin(i))? This answer may be written as P(sin(i) < Z|M₀) and also depends on P(M_T).

To relate the sin(i) probability distribution and the true mass distribution, it is simply

$$\begin{equation} P(M_T < X | M_0) = 1-P(\sin (i) < Z | M_0) \end{equation} \tag{ 3 }$$

for $Z = \frac{M_0}{X}$.

To illustrate the fundamental issue, we consider the following toy model: suppose that all exoplanets have a true mass of either 1.0 M_J or 2.0 M_J, where M_J is the mass of Jupiter and that there are an equal number of exoplanets with each of these masses. If an exoplanet is determined to have M₀ = M_T sin(i) = 0.5 M_J, the value of sin(i) is obviously either 0.5 or 0.25 depending on whether it is one of the low or high true mass exoplanets, respectively. Moreover, since a sin(i) value of 0.5 is about 2.236 times more likely than one of 0.25 for the prior (isotropic i) distribution of sin(i), it follows that the posterior distribution of sin(i) for this system consists of two δ functions, one at 0.5 and one at 0.25 with the former having an amplitude 2.236 times that of the latter.

Since any intrinsic exoplanet mass distribution could be arbitrarily well approximated by a series of δ functions, we can conclude that the intrinsic mass distribution affects the posterior distribution of the sin(i), for any particular observed value of M₀.

Consider a planet at any mass M_T. Given that the inclination angle i is randomly (isotropically) distributed, we know that (from the previous sections) the pdf of sin(i) falling into a range of sin(i) − d sin(i) and sin(i) + d sin(i) is $\sin (i)/\sqrt{1-\sin ^2(i)}$. The culmulative probability of sin(i) < Z will then be

$$\begin{equation} P(\sin (i)< Z) = \int _0^Z \sin (i)/\sqrt{1-\sin ^2(i)}\, d\sin (i), \end{equation} \tag{ 4 }$$

which is simply

$$\begin{equation} P(\sin (i) <Z) = 1- \cos (\arcsin (Z)). \end{equation} \tag{ 5 }$$

This is however not surprising, since we know (from trigonometric argument) that

$$\begin{equation} P(i<x) = 1-\cos (x), \end{equation} \tag{ 6 }$$

and therefore, we know that the prior probability of finding sin(i) less than Z, which is equivalent to the prior probability of observed mass M₀ given M_T is just

$$\begin{equation} P(\sin (i) < Z) = 1- \cos (\arcsin (Z)). \end{equation} \tag{ 7 }$$

This, of course, is simply the prior pdf of sin(i), derived in Section 1. This distribution function is plotted in the upper panel of Figure 1 in cumulative form; we can also look at the probability of observed mass M₀ given M_T, which is simply the following:

$$\begin{equation} P(M_0 | M_T) = \frac{\big(M_0/M_T^2\big)}{\sqrt{1-\big(\frac{M_0}{M_T}\big)^2}}. \end{equation} \tag{ 8 }$$

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Thus, the culmulative prior probability of observed mass M₀ given M_T (note that we are here fixing the true mass M_T and calculating the prior distribution of the observed mass M₀, while in the analyses which follow we will do just the reverse to obtain posterior distributions) and setting M_T = 1 is just

$$\begin{equation} P(M_0<X | M_T=1) = 1-\sqrt{1-X^2}. \end{equation} \tag{ 9 }$$

As expected, the plot has the same behavior as in P(sin(i) < Z), as plotted in the lower panel of Figure 1.

Proceeding now to posterior distributions, Bayes' theorem states

$$\begin{equation} P(A|B) = \frac{P(B|A) P(A)}{P(B)}. \end{equation} \tag{ 10 }$$

Assigning A = M_T and B = M₀, we directly obtain

$$\begin{equation} P(M_T | M_0) = \frac{P(M_0 | M_T) P(M_T) }{P(M_0)}. \end{equation} \tag{ 11 }$$

Equation (8) gives the first term in the numerator of Equation (11). The second term in the numerator is an unknown function (which ultimately may be determined from observations), but it is possible to consider simple toy models, plausible guesses, and even theoretical estimates for P(M_T) and thus explore their consequences. Finally, the denominator of Equation (11) can be obtained from

$$\begin{equation} P(M_0) = \int P(M_0 | M_T) P(M_T)\, dM_T. \end{equation} \tag{ 12 }$$

Therefore,

$$\begin{equation} P(M_T| M_0) = \frac{P(M_0 | M_T) P(M_T)}{ \int P(M_0 | M_T) P(M_T)\, dM_T }, \end{equation} \tag{ 13 }$$

which provides the desired posterior distribution

$$\begin{equation} P(M_T| M_0) = \frac{\frac{(M_0/M_T^2)}{\sqrt{1-\big(\frac{M_0}{M_T}\big)^2}} P(M_T)}{ \int \frac{(M_0/M_T^2)}{\sqrt{1-\big(\frac{M_0}{M_T}\big)^2}} P(M_T)\, dM_T } \end{equation} \tag{ 14 }$$

for M₀ < M_T.

It is frequently most interesting to consider instead the cumulative probability distribution at which M_T < X, requiring the integration of the numerator of Equation (13) up to X. This gives

$$\begin{equation} P(M_T < X | M_0) = \frac{\int ^X\, P(M_T| M_0) P(M_T)\, dM_T}{ \int\, P(M_T| M_0) P(M_T)\, dM_T }. \end{equation} \tag{ 15 }$$

Since if the observed mass is M₀, then the true mass M_T has to be larger than or equal to M₀, since M₀ = M_T sin(i) and sin(i) ⩽ 1, therefore the lower integral limit is M₀, and the upper mass limit could be as large as physically possible for the mass of a planet (M_max).

Therefore, we have the following:

$$\begin{equation} P(M_T < X | M_0) = \frac{\int _{M_0}^X \frac{(M_0/M_T^2)}{\sqrt{1-\big(\frac{M_0}{M_T}\big)^2}} P(M_T)\, dM_T}{ \int _{M_0}^{M_{{\rm max}}} \frac{(M_0/M_T^2)}{\sqrt{1-\big(\frac{M_0}{M_T}\big)^2}} P(M_T)\, dM_T }. \end{equation} \tag{ 16 }$$

The upper bound on the integral in the denominator M_max is somewhat arbitrary, corresponding to the maximum mass of any planet drawn from the P(M_T) distribution. However, the value of M_max affects only the normalization of P(M_T < X|M₀), not its form.

This formulation in terms of M_T most transparently displays the underlying logic of the derivation. However, the same approach can give equally well the answer to question No. 2 above, since the two are equivalent. In particular,

$$\begin{equation} P(\sin (i) < Z | M_0) = P(M_T > X | M_0). \end{equation} \tag{ 17 }$$

Thus,

$$\begin{equation} P(\sin (i) < Z | M_0) = 1- \frac{\int _{M_0}^{M_0/Z} \frac{(M_0/M_T^2)}{\sqrt{1-\big(\frac{M_0}{M_T}\big)^2}} f_{M_T}(y)\,dy}{\int _{M_0}^{M_{{\rm max}}}\frac{(M_0/M_T^2)}{\sqrt{1-\big(\frac{M_0}{M_T}\big)^2}} f_{M_T}(y)\, dy}, \end{equation} \tag{ 18 }$$

where $f_{M_T}(y)$ is the true mass distribution and y is a dummy variable of integration. The normalization of $f_{M_T}$ cancels out of the expression.

Due to our current ignorance of the true P(M_T), we cannot evaluate these expressions uniquely for the actual observed values of M₀ of known exoplanets. It is nevertheless instructive to do so for various assumed P(M_T) distributions. We devote the remainder of the paper primarily to that exercise.

In order to investigate the size and character of the statistical effect under discussion, we will apply the formula derived in the previous section to a series of simple "toy models" of P(M_T), some of which might turn out to reassemble reality at least qualitatively.

In other words, the specific models considered in the subsections below are not motivated by what is currently known about the M_T sin(i) from RV observations but rather have been chosen simply to illustrate the nature and theoretically possible extent of the statistical effect under consideration. Nevertheless, some of the models are at least qualitatively similar to the M_T distribution suggested by RV data for exoplanets with M_T sin(i) values greater than one or a few tenths of Jupiter's mass. Others are not but might well be qualitatively representative of exoplanet populations which are as yet little or not at all characterized by RV data. For example, theoretical scenarios for exoplanet formation (Ida & Lin 2008a, 2008b; Matsumura et al. 2009; Baraffe et al. 2010 and references therein) predict a significant "gap," a second "desert" in effect, in the mass distribution in the range of a few tens of Earth masses. If this feature actually exists, there must be a mass range in which the exoplanet mass function increases with increasing mass, just the opposite of the sharp decrease with the M_T sin(i) distribution seen in the mass range characterized well by RV studies to date. Thus, some of the "toy models" discussed in this section might be qualitatively relevant for a mass range near such a "gap" even if they are definitely irrelevant for the currently well-studied one.

4.1. Power-law M_T Distributions

Beginning with a particularly simple possibility, we now assume that the distribution of true masses of exoplanets follow a power law, thus we adopt the form $f_{M_T}(y) = A y^\alpha$, where both A and α are constants. Then we can evaluate the main integral (hereafter $\Phi (M_0, \alpha) = \int _{M_0}^X \frac{(M_0/M_T^2)}{\sqrt{1-(\frac{M_0}{M_T})^2}} A M_T^\alpha dM_T$) of Equation (16) for several cases (α = 2, α = 1, α = 0, α = −1, α = −2) to obtain the following:

$$\begin{equation} \Phi (M_0,X,\alpha) = \left\lbrace \begin{array}{@{}ll@{}} \sqrt{X^2-1} & {\rm if \; } \alpha = 2, \\ \ms \log (2\sqrt{X^2-1}+ X) & {\rm if \; } \alpha = 1, \\ \ms \pi /2 -\tan ^{-1}\big(\frac{1}{\sqrt{X^2 -1}}\big) & {\rm if \; } \alpha = 0, \\ \ms \sqrt{1-\frac{1}{X^2}} & {\rm if \; } \alpha = -1,\\ \ms \frac{X^2-X^2\sqrt{X^2-1}\tan ^{-1}\big(\frac{1}{\sqrt{X^2-1}}\big)-1}{2X^3\sqrt{1-1/X^2}} & {\rm if \; } \alpha = -2. \end{array} \right. \end{equation} \tag{ 19 }$$

Note that we set M₀ = 1 and A = 1 to simplify the equations; this is equivalent to a simple change in the units of mass and space density.

For the distribution of sin(i), we can refer to Equation (3). We use this result to plot a few specific cases in Figure 2, assuming various values of α.

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Note from Figure 2 that α = −1 gives a median sin(i) value of 0.860 while an equally plausible value of α = 0 gives a median sin(i) value of 0.704. If the mass distribution is an increasing function of mass, the resulting median sin(i) value will be reduced quite dramatically; for example, α = 2 gives a median sin(i) value of 0.02.

It is equally easy to generate the corresponding P(M_T < X|M₀) distributions, using Equation (16), as shown in Figure 3. Since the observed mass is set to 1, the true mass has to be larger than 1, and as the power-law index increases (which means a larger number of high-mass planets in the true mass distribution), the probability of finding a planet below X decreases (as seen in Figure 3).

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4.2. Power Law plus a Delta Function M_T Distributions

The distribution of planetary masses in the solar system, the highly nonlinear and at least partially non-gravitational nature of planet formation as well as some specific theoretical models (see Kokubo & Ida 1996; Kokubo et al. 2006; Ida & Lin 2008a; Baraffe et al. 2010 and references therein) suggest that the P(M_T) distribution might contain one or more characteristic masses, rather than being an entirely scale-free power law. In order to investigate the implications of such a P(M_T), we consider a toy model in which some of the exoplanets are distributed in an α = −1 power-law population while the others all have the same mass M_c. We may then again evaluate the expressions of Section 2 directly.

Thus, we have $f_{M_T}(y)= A y^\alpha + B \delta (y-M_c)$, where M_c is the critical mass scale of interest. It is convenient to introduce the dimensionless parameter η, defined by M₀ = ηM_c, and to set A and B equal. Without loss of generality M_c = 1 is adopted (i.e., M_c is defined as the unit of mass, rather than M₀ as in the pure power-law models analyzed in the previous subsection) for purposes of plotting and giving numerical values. We can then obtain the following:

$$\begin{equation} P(M_T <X | \eta) = \frac{ \frac{1}{\eta }\sqrt{1-\frac{\eta ^2}{X^2}} + \frac{\eta }{\sqrt{1-\eta ^2}} }{\frac{1}{\eta }\sqrt{1-\frac{\eta ^2}{M_{{\rm max}}^2}} + \frac{\eta }{\sqrt{1-\eta ^2}}} \end{equation} \tag{ 20 }$$

given that M₀ < M_c < X < M_max. It is easy to see that the addition of $\frac{\eta }{\sqrt{1-\eta ^2}}$ will increase the probability that M_T is smaller than X.

Furthermore, if M₀ < X < M_c < M_max, we have

$$\begin{equation} P(M_T <X | \eta) = \frac{ \frac{1}{\eta }\sqrt{1-\frac{\eta ^2}{X^2}} }{\frac{1}{\eta }\sqrt{1-\frac{\eta ^2}{M_{{\rm max}}^2}} + \frac{\eta }{\sqrt{1-\eta ^2}}}. \end{equation} \tag{ 21 }$$

This makes sense as the critical mass scale is not within the boundary that we consider (M_T < X), so the probability decreases.

Finally, if M₀ < X < M_max < M_c, then the results are similar to the original situation when $f_{M_T}(y) = Ay^{\alpha }$ except that some of the planets are in the δ function part of the distribution, thus reducing the relative probability of sampling the power-law portion:

$$\begin{equation} P(M_T <X | \eta) = \frac{1}{2} \frac{ \frac{1}{\eta }\sqrt{1-\frac{\eta ^2}{X^2}} }{\frac{1}{\eta }\sqrt{1-\frac{\eta ^2}{M_{{\rm max}}^2}}}. \end{equation} \tag{ 22 }$$

We can also obtain the distribution of sin(i):

$$\begin{equation} P(\sin (i) < Z | \eta) = 1-\frac{1}{2} \frac{ \frac{1}{\eta }\sqrt{1-Z^2} }{\frac{1}{\eta }\sqrt{1-\frac{\eta ^2}{M_{{\rm max}}^2}}}, \end{equation} \tag{ 23 }$$

if 0 < sin(i) < M₀/M_c < M₀/X.

If we set $Z=\frac{M_0}{X}$ and M₀ = ηM_c, while M_c = 1, then we can plot Figure 4. It illustrates the discontinuity in the probability at the delta function (i.e., when η = Z).

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4.3. A Solar System Like Mass Distribution

Turning now to a more complex but also more physically plausible distribution, we analyze the case of exoplanet masses distributed in a way similar to that of solar system planets. This distribution can be modeled very roughly as two power laws separated by a gap in mass. One power law lies at a low-mass range (the terrestrial planets) while the other lies at a much higher mass range (the giant planets). We consider a toy model with two power-law mass distributions, one extending from 1M_c to 20M_c, while the other power law is for 400M_c to 8000M_c. There are no planets in the range between 20M_c and 400M_c. We also assume that the two power laws have the same power index, and also the same coefficient (i.e., $f_{M_T}(y) = A y ^\alpha$ in range of 1M_c to 20M_c and $f_{M_T}(y) =B y^\beta$ in the range of 400M_c to 8000M_c, where A = B and α = β). We plot the probability P(sin(i) < Z|η) as η varies (the ratio of the observed mass M₀ to the critical mass M_c) for α = −1 in Figure 5. Note that the probability P(sin(i) < Z|η) can saturate very near either unity or zero over a substantial range of Z values depending on the value of η.

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In the preceding analysis we have consistently assumed that exoplanets discovered by the RV method uniformly (i.e., without bias) sample the distribution of M_T and sin(i) values in nature. Obviously, this is unrealistic. In reality, both variables (and others) influence the probability that a given exoplanet system will be detected in an RV survey, and this selection bias in turn affects the likely values of both M_T and sin(i).

Happily, this complication does not fundamentally alter our results because the basic effect discussed in this paper is a purely statistical one, independent of any observational biases. More specifically, one could conduct an exactly parallel analysis in which the true distributions of M_T and sin(i), which appear in Equations (16)–(18), are replaced with the biased distributions which a particular RV survey samples, if its selection function can be determined reasonably accurately.

A very simple example would be a case in which the probability of an RV survey detecting an exoplanet of mass M_T is given by some selection function S(M_T), independent of sin(i) and other properties of the system. In that case, it suffices to replace P(M_T) with S(M_T)P(M_T) everywhere it occurs in the equations and proceed as before.

The primary implication of the results presented here is that in general the value of sin(i) for a given exoplanet system will not be drawn from its prior distribution, corresponding to an isotropic distribution of i as is often assumed, at least implicitly.

The relevant, i.e., posterior, probability distribution of sin(i) depends sensitively on the distribution of true masses M_T and the observed mass M₀ = M_T sin(i). Since the former is not well constrained, either empirically or theoretically, at present the true mass M_T of such a system cannot be trivially estimated from the value of M₀ as is also often assumed to be the case (see Butler et al. 2004; Mayor et al. 2005, 2009; Lovis et al. 2006; Wright et al. 2008).

This means, for example, that it is difficult to identify the least (or most) massive RV exoplanets discovered to date because selecting low values of M_T sin(i) from an observed exoplanet sample is a way of picking out low sin(i) values as well as low M_T values. For some possible exoplanet mass distributions the observed objects with the lowest observed M_T sin(i) will be dominated by systems with small sin(i) values rather than small masses.

It also implies that the distribution of true exoplanet masses is not necessarily similar in shape to the distribution of M₀, even for large samples, as is sometimes taken to be the case (see Mayor et al. 2005; Butler et al. 2006; Cumming et al. 2008). Such an approximation may be valid for some exoplanet populations or mass ranges but quite misleading for others.

The moral of the above analysis is that the sin(i) factor should be given its due in RV exoplanet studies. For example, we urge that RV observers reporting the value of M_T sin(i), typically for a newly discovered planet, also report a confidence interval for M_T at some standard selected level (e.g., 95%) based on some explicitly stated assumption for the true exoplanet mass distribution.

As a simple illustration, the 95% confidence intervals for M_T if M₀ = 1.0 are 1.0001–2.405, 1.0017–4.566, 1.005–27.02, 1.15–85.186, and 1.125–94.34 for the simple power-law M_T distributions considered in Section 4.1 with assumed power-law slopes of α = −2, −1, 0, +1, and +2, respectively. Note that the α = −1 case is equivalent to that of an isotropic i distribution; in other words, if the exoplanet true mass distribution is uniform in logarithmic intervals, the posterior sin(i) distribution is the same as the prior one. Moreover, if α is less than −1, M_T values will be even closer to M₀ values than in the isotropic i distribution case. This means that ignoring the issue raised in this paper can lead one to over, as well as under, estimate mass uncertainties. To illustrate this dependence better, we plot the probability distribution function of the posterior probability distribution of sin(i) assuming various values of slope α in Figure 6.

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Although these model-dependent upper bounds may appear less impressive or exciting than the M_T sin(i) value itself, they are a less misleading, and thus more scientifically informative indication of the actual information on any particular exoplanet's mass provided by RV data alone.

Although this paper concerns itself only with the estimation of true masses of individual exoplanets on the basis of RV data, it is clearly relevant to the more challenging and, in some respects, more fundamental problem of estimating the distribution of true masses for various exoplanet populations. The classic work of Chandrasekhar & Münch (1950) long ago demonstrated that problems of this form are non-trivial, in particular that straightforward inversions of the data are unstable. Modern studies (Jorissen et al. 2001; Zucker & Mazeh 2001; Marcy et al. 2005; Udry & Santos 2007; Brown 2011) have addressed the issue with a variety of mathematical approaches and assumptions, some relatively sophisticated, but it is likely that considerably more work will be required before the exoplanet mass distribution(s) can be regarded as well determined. Until that goal is accomplished, RV mass determinations of individual exoplanets will be subject to the systematic sin(i) uncertainty described in this paper.

We thank Dan Fabrycky, Scott Gaudi, John Johnson, Geoff Marcy, Tim Morton, David Spergel, Dave Spiegel, and Jason Wright for useful comments and suggestions. S.H. acknowledges support from the Lawrence Berkeley National Laboratory Seaborg Fellowship and Chamberlain Fellowship and support from the Princeton University Department of Astrophysics as S.H. started this project when she was a graduate student at Princeton University. E.L.T. gratefully acknowledges support from a Princeton University Global Collaborative Research Fund grant and the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. E.L.T. also acknowledges the support of the Research Center for the Early Universe (RESCEU) at the University of Tokyo and the hospitality of its Department of Physics.