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The mass of the young planet Pictoris b through the astrometric motion of its host star

The young massive Jupiters discovered with high-contrast imaging provide a unique opportunity to study the formation and early evolution of gas giant planets. A key question is to what extent gravitational energy from accreted gas contributes to the internal energy of a newly formed planet. This has led to a range of formation scenarios from 'cold' to 'hot' start models. For a planet of a given mass, these initial conditions govern its subsequent evolution in luminosity and radius. Except for upper limits from radial velocity studies, disk modelling, and dynamical instability arguments, no mass measurements of young planets are yet available to distinguish between these different models. Here we report on the detection of the astrometric motion of Beta Pictoris, the 21 Myr-old host star of an archetypical directly-imaged gas giant planet, around the system's centre of mass. Subtracting the highly accurate Hipparcos-Gaia proper motion from the internal 3-yr Hipparcos astrometric data reveals the reflex motion of the star, giving a model-independent planet mass of M=11+-2 MJup. This is consistent with scenarios in which the planet is formed in a high-entropy state as assumed by hot start models. The ongoing data collection by Gaia will in the near future lead to mass measurements of other young gas giants and form a great asset to further constrain early evolution scenarios.

typical uncertainties in the range 1.5 -2.5 mas, the subsequent reprocessing of this data by van Leeuwen 12 maintained the individual data points, and delivered uncertainties in the range 0.8-1.0 mas (see Supplementary Table 1), improving the precision by a factor of ~4. The latter analysis provides for Beta Pictoris a parallax of 51.44±0.12 mas, corresponding to a distance of 19.44±0.05 pc, and a proper motion of 4.65±0.11 mas yr -1 and 83.10±0.15 mas yr -1 in the right ascension and declination directions ( Figure 1; Table 1). The star is the largest member of an association of young stars, the Beta Pictoris moving group (BPMG), sharing the same origin and motion through space.
Beta Pictoris is included in the second data release 16   Telescope 19-21 . It means that while about half of the orbit is well known, the orbital period is still poorly constrained to be between 20.2 to 26.3 years (ref 19). Since the reference epochs of Hipparcos (1991.25) and Gaia DR2 (2015.5) are separated by 24.25 years, the observations of the former mission fall in the well-determined part of the planet orbit, but the precise orbital phase is uncertain. Depending on the orbital period, the Hipparcos-Gaia baseline also contains a small part of the stellar reflex motion. E.g. for an orbital period of 22 years and planet mass of 10 M Jup , the two reference epochs correspond to orbital phases that differ by 0.1, resulting in a planet-induced position difference of 1.0 mas and an effect on the Hipparcos-Gaia baseline of 1.0/24.25 = 0.04 mas yr -1 . This effect is taken into account in our calculations below.
Since the projected orientation of the orbit is well constrained 19 , so is the expected direction of stellar reflex motion. Hence, we converted the positional data from Hipparcos to onedimensional position measurements in this direction as function of time, which is shown in the right panel of Figure 2. The solid curves show the best-fit stellar motion for several trial orbital periods, indicating that short periods are not consistent with the Hipparcos data since this would result in significant acceleration in the stellar proper motion in the period 1990-1993, which is not observed. Figure 3 shows the constraints to the planet mass and orbital period. The greenshaded area indicates the 1s uncertainty interval on the orbital period from the high-contrast imaging monitoring 19 , and the contours the 1, 2, and 3s limits on orbital period and mass from the study presented here. The planet mass is constrained to 11±2 M Jup (1s), and the orbital period is likely to be >22.2 years (2s limit). Previous upper limits to the planet mass of <20-30 M Jup from radial velocity monitoring 9,10 are consistent with the astrometric measurement presented here.
We compared our measurement with the mass estimates obtained from gas giant cooling models. The wide range of estimates in the literature are driven by different assumptions on the age of the planet, and the internal energy of the planet at time of formation. In particular, it is uncertain what fraction of the released gravitational energy of newly accreted material during formation is lost in the accretion shock on the planet surface or circumplanetary disk and not incorporated in the planetary structure, resulting in a range of so called "hot" start, "cold" start, and intermediate "warm" start models 5-8 . In addition, different age-determination methods give different stellar ages, and also it is not certain whether the planet was formed at exactly the same time as the star, or possibly a few millions of years later. Isochronal fits to low-mass BPMG members provide 22  in line with the model-independent measurement presented here, implying that the underlying assumptions in the hot start models and the age estimates of the system are largely correct.
The ongoing data collection by Gaia will in the near future lead to a further improvement on the mass measurement of Beta Pictoris b by up to an order of magnitude, and the masses of other known young planets will be in reach 17,18 . This means that Gaia will form a great asset to further constrain early evolution scenarios. In addition, Gaia is poised 29 to find many new extrasolar planets.

Acknowledgments
This work was only possible because numerous scientists and engineers have devoted large parts of their careers to the design, construction, and successful operation of the Hipparcos and Gaia missions, and the analysis of their data. We are grateful. We thank Floor van Leeuwen also for discussions on his analysis of the Hipparcos data. This work has made use of data from

Table 1
The  show significantly more curvature than the Hipparcos data allows.

Methods: Data Analysis
We collected the Hipparcos intermediate data products from the van Leeuwen 2007 reduction 14 from CD, which contain for each observation the spacecraft orbit and epoch, the cosine and sine of the scan orientation, the residual to the best fit solution, and the formal error (see Supplementary Table 1). All calculations were performed with custom-built IDL scripts. Since the direction of the reflex motion of the host star is well determined 19 (j = 212 o ), we converted each orbit-averaged observation into a position-measurement along this direction and adjusted its associated uncertainty, which depends on 1/cosine of the angle relative to the scanning direction. This is shown in the right panel of Figure 2, where the data points have grey scales depending on their uncertainty to emphasis the data with the smallest errors. Five measurements were omitted because their scanning directions were near-perpendicular leading to very large errors >10 mas. The shape of the orbit of the planet is well determined for the Hipparcos 1990-1993 observing interval, because it is approximately one planet orbit (20 -26 yrs) before the current era (2010 -present) during which the planet position has been frequently monitored (see Supplementary Figure 1). Therefore, although the orbital period is not well constrained, the best-fit 19 solution to its orbit can be used to fit the reflex motion of the star, modulo a scaling depending on the planet/star mass ratio and a phase shift governed by the assumed orbital period. This is shown for trial orbital periods between 21 and 24 years in the right panel of Figure 2. For a short orbital period of 21 years, the Hipparcos observations fall around the epoch of longest elongation, implying a strong acceleration/deceleration of its angular motion -which is excluded by the Hipparcos data.
We subsequently performed a chi-squared analysis to the data presented in the right panel of Data availability. The data that support the plots within this paper and other findings are available in Table 1