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arXiv 1007.0245astro-ph.EP
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由径向速度数据确定M_Tsin(i)系外行星sin(i)的后验分布

The Posterior Distribution of sin(i) for Exoplanets with M_T sin(i) Determined from Radial Velocity Data

Shirley Ho, Edwin L. Turner

摘要:

给出了一个系外行星系统的径向速度观测值M_Tsin(i)条件(即:不仅给出行星的真实质量M_T的信息,而且给出该系统的sin(i)的值(其中i是轨道倾角)。因此,对于具有任何特定的M_T sin(i)观测值的系统,sin(i)的值不能被假设为从对应于各向同性i分布的分布(即假定的先验分布)中随机抽取。相反,后验分布取决于所研究的系外行星种群的M_T的内在分布。我们给出了一个简单的贝叶斯推导这种关系,并将其应用到几个“玩具模型”的(目前未知的)内在分布的M_T。结果表明,这种影响可能是一个重要的影响。例如,即使对于M_T的简单幂律分布,对于幂律指数(α)在-2和+1之间的范围内,观察到的RV样本中sin(i)的中值可以分别在0.860和0.023之间变化(与各向同性i分布的0.866值相比)。在相同的指标范围内,由于sin(i)的不确定性,M_T的95%置信区间比M_T sin(i)大1.002-4.566(α = -2)至1.13-94.34(α = +1)倍。我们的定性结论是,RV研究的系外行星,无论是个人的对象和统计样本,应该把sin(i)的因素比一个简单的和很好理解的统计特性的“数字常数的顺序单位”。我们认为,为了更准确地反映与RV研究相关的质量不确定性,M_T sin(i)测定的报告应附有M_T在95%水平上的相应置信限的声明,该置信限是基于明确声明的真实M_T分布的假定形式。

英文摘要:

Radial velocity (RV) observations of an exoplanet system giving a value of M_T sin(i) condition (ie. give information about) not only the planet's true mass M_T but also the value of sin(i) for that system (where i is the orbital inclination angle). Thus the value of sin(i) for a system with any particular observed value of M_T sin(i) cannot be assumed to be drawn randomly from a distribution corresponding to an isotropic i distribution, i.e. the presumptive prior distribution . Rather, the posterior distribution from which it is drawn depends on the intrinsic distribution of M_T for the exoplanet population being studied. We give a simple Bayesian derivation of this relationship and apply it to several "toy models" for the (currently unknown) intrinsic distribution of M_T. The results show that the effect can be an important one. For example, even for simple power-law distributions of M_T, the median value of sin(i) in an observed RV sample can vary between 0.860 and 0.023 (as compared to the 0.866 value for an isotropic i distribution) for indices (alpha) of the power-law in the range between -2 and +1, respectively. Over the same range of indicies, the 95% confidence interval on M_T varies from 1.002-4.566 (alpha = -2) to 1.13-94.34 (alpha = +1) times larger than M_T sin(i) due to sin(i) uncertainty alone. Our qualitative conclusion is that RV studies of exoplanets, both individual objects and statistical samples, should regard the sin(i) factor as more than a "numerical constant of order unity" with simple and well understood statistical properties. We argue that reports of M_T sin(i) determinations should be accompanied by a statement of the corresponding confidence bounds on M_T at, say, the 95% level based on an explicitly stated assumed form of the true M_T distribution in order to more accurately reflect the mass uncertainties associated with RV studies.

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