We estimated the upper limit masses of an extrasolar planetary system, υ And. Doppler observation of extrasolar planets through the radial velocity displacement of their host stars can determine only lower limits of planetary masses and inevitably involves uncertainties due to the unknown line-of-sight inclination of planetary orbits, 1/ sin i. However, this inherent uncertainty about mass determination may be compensated by long-term numerical integrations of planetary orbital motions. Our numerical integrations, though our description is limited to only two outer companions around υ And, intend to find any relation between the planetary masses and their dynamical timescales of instability. We integrate the planetary orbits for the Gyr time span with various initial conditions of masses and angle variables to investigate which kind of initial configuration produces stable orbits during the timescale of the υ And host star's age. According to numerical results starting from the Lick Observatory data set, the υ And planetary system is likely to remain stable over the timescale of its host star's age if sin i ≥ 0.7. In this case we may estimate that the upper limit masses of the υ And planets in our model are about 1/0.7 ~ 1.43 times larger than their minimum (i.e., when sin i = 1). However, the instability timescale of the planetary system depends much on its initial orbital configuration due to its chaotic character. Another interesting feature in the orbital motion of the υ And planets is that the longitudes of pericenter of the outer two companions are likely to align, which stabilizes the system. Systems where this orbital alignment is not well achieved easily become unstable.
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机译：我们估算了太阳系外行星系统υAnd的上限质量。通过其主恒星的径向速度位移对太阳系外行星进行多普勒观测只能确定行星质量的下限，并且由于行星轨道的视线倾角未知，1 /不可避免地会带来不确定性。但是，这种有关质量确定的内在不确定性可以通过行星轨道运动的长期数值积分得到补偿。尽管我们的描述仅限于υAnd附近的两个外部同伴，但我们的数值积分旨在寻找行星质量与其动力学不稳定性时标之间的任何关系。我们将Gyr时间跨度的行星轨道与质量和角度变量的各种初始条件进行了整合，以研究在υ和宿主恒星年龄范围内哪种初始构型会产生稳定的轨道。根据从里克天文台数据集开始的数值结果，如果sin i≥0.7，则υ和行星系统可能会在其宿主恒星年龄的时间范围内保持稳定。在这种情况下，我们可以估计模型中υ和行星的上限质量约为其最小值的1 / 0.7〜1.43倍（即，当sin i = 1时）。然而，由于其混沌特性，行星系统的不稳定性时标在很大程度上取决于其初始轨道构型。 υ和行星的轨道运动的另一个有趣特征是外两个同伴的中心点的经度可能会对齐，从而使系统稳定。不能很好地实现轨道对准的系统会变得不稳定。
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