Astrometric exomoon detection by means of optical interferometry

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Table 1

Astrometric signal amplitudes for different case examples in the top and the approximate astrometric measurement precision of different current and future instruments in the bottom.

Planet	System parameters			M_m (M_Jup)	χ	A (μas)
β Pic b	ϖ = (50.93 ± 0.15) mas ¹			1.0	0.5	3000 ± 600
	M_s = (1.83 ± 0.04) M_⊙ ²			1.0	0.1	600 ± 130
	M_pl = ${9.3}_{- 2.5}^{+ 2.6}$ $\[9.3_{-2.5}^{+2.6}\]$ M_Jup ²			0.1	0.5	330 ± 70
				0.1	0.1	65 ± 15
	a_pl = (10.26 ± 0.10) AU ²			0.01	0.5	33 ± 8
	R_{Hill, pl} = (1.20 ± 0.11) AU			0.01	0.1	6.6 ± 1.5

HR 8799 d	ϖ = (24.46 ± 0.05) mas ¹			1.0	0.5	3800 ± 900
	M_s = ${1.51}_{- 0.24}^{+ 0.38}$ $\[1.51_{-0.24}^{+0.38}\]$ M_⊙ ³			1.0	0.1	770 ± 180
	M_pl = (10 ± 3) M_Jup ⁴			0.1	0.5	420 ± 110
	M_pl = (10 ± 3) M_Jup ⁴			0.1	0.1	80 ± 20
	a_pl = (26.97 ± 0.73) AU ³			00.1	0.5	42 ± 11
	R_{Hill, pl} = (3.5 ± 0.4) AU			00.1	0.1	8 ± 2

HD 206893 c	ϖ = (24.53 ± 0.04) mas ¹			1.0	0.5	460 ± 30
	M_s = ${1.32}_{- 0.05}^{+ 0.07}$ $\[1.32_{-0.05}^{+0.07}\]$ M_⊙ ⁵			1.0	0.1	92 ± 7
	M_pl = ${12.7}_{- 1.0}^{+ 1.2}$ $\[12.7_{-1.0}^{+1.2}\]$ M_Jup ⁵			0.1	0.5	49 ± 4
				0.1	0.1	9.8 ± 0.7
	a_pl = ${3.53}_{- 0.06}^{+ 0.08}$ $\[3.53_{-0.06}^{+0.08}\]$ AU ⁵			00.1	0.5	5.0 ± 0.4
	R_{Hill, pl} = (0.51 ± 0.02) AU			00.1	0.1	0.99 ± 0.07

	SPHERE	GPI	GRAVITY	PLANETES*	MICADO^†	KBI^††

σ_A (μas)	1500 ⁶	1000 ⁷	50 ⁸	10 ⁹	400 ⁹	< 1 ¹⁰

Notes. In the top table, ϖ, M_s, M_pl, and a_s are the system’s parallax, stellar mass, planetary mass, and semi-major axis, respectively. R_{Hill, pl} is the Hill radius that results from the system parameters according to Equation (2) when propagating the uncertainties associated with the different quantities involved. The semi-major axis of the moon used to compute the signal amplitude, A, via Equation (1), is the respective Hill radius scaled by the constant fraction, χ. Again, all uncertainties were propagated until arriving at the signal amplitude. In the bottom table, σ_A is the approximate astrometric measurement uncertainty achievable with different instruments. * Future VLTI instrument (see Sect. 3); ^† First-generation ELT instrument (Davies et al. 2021); ^†† Future kilometre-baseline interferometric facility (Bourdarot & Eisenhauer 2024); ¹ Gaia Collaboration (2023); ² Brandt et al. (2021); ³ Zurlo et al. (2016); ⁴ Marois et al. (2008); ⁵ Hinkley et al. (2023); ⁶ Maire et al. (2021); ⁷ Wang et al. (2016); ⁸ GRAVITY Collaboration (2019); ⁹ priv. comm.; ¹⁰ Bourdarot & Eisenhauer (2024).

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