Galactic center S-star orbital parameters

ida (") Δa e Δe i (°) Δi Ω (°) ΔΩ ω (°) Δω Tp (yr) ΔTp P (yr) ΔP Kmag r (AU) Δr v (%c) Δv
S1 0.5950 0.0240 0.5560 0.0180 119.14 0.21 342.04 0.32 122.30 1.40 2001.800 0.150 166.0 5.8 14.70 2160.7 6.7 0.55 0.03
S2 0.1251 0.0001 0.8843 0.0001 133.91 0.05 228.07 0.04 66.25 0.04 2018.379 0.001 16.1 0.0 13.95 118.4 0.2 2.56 0.00
S4 0.3570 0.0037 0.3905 0.0059 80.33 0.08 258.84 0.07 290.80 1.50 1957.400 1.200 77.0 1.0 14.40 1779.7 25.1 0.57 0.01
S6 0.6574 0.0006 0.8400 0.0003 87.24 0.06 85.07 0.12 116.23 0.07 2108.610 0.030 192.0 0.2 15.40 860.3 4.4 0.94 0.00
S8 0.4047 0.0014 0.8031 0.0075 74.37 0.30 315.43 0.19 346.70 0.41 1983.640 0.240 92.9 0.4 14.50 651.7 22.5 1.07 0.01
S9 0.2724 0.0041 0.6440 0.0200 82.41 0.24 156.60 0.10 150.60 1.00 1976.710 0.920 51.3 0.7 15.10 793.2 36.9 0.93 0.02
S12 0.2987 0.0018 0.8883 0.0017 33.56 0.49 230.10 1.80 317.90 1.50 1995.590 0.040 58.9 0.2 15.50 272.9 2.0 1.69 0.01
S13 0.2641 0.0016 0.4250 0.0023 24.70 0.48 74.50 1.70 245.20 2.40 2004.860 0.040 49.0 0.1 15.80 1242.0 2.4 0.69 0.01
S14 0.2863 0.0036 0.9761 0.0037 100.59 0.87 226.38 0.64 334.59 0.87 2000.120 0.060 55.3 0.5 15.70 56.0 3.8 3.83 0.06
S17 0.3559 0.0096 0.3970 0.0110 96.83 0.11 191.62 0.21 326.00 1.90 1991.190 0.410 76.6 1.0 15.30 1755.3 16.4 0.57 0.02
S18 0.2379 0.0015 0.4710 0.0120 110.67 0.18 49.11 0.18 349.46 0.66 1993.860 0.160 41.9 0.2 16.70 1029.3 3.8 0.77 0.01
S19 0.5200 0.0940 0.7500 0.0430 71.96 0.35 344.60 0.62 155.20 2.30 2005.390 0.160 135.0 14.0 16.00 1063.3 4.5 0.83 0.20
S21 0.2190 0.0017 0.7640 0.0140 58.80 1.00 259.64 0.62 166.40 1.10 2027.400 0.170 37.0 0.3 16.90 422.7 3.6 1.32 0.02
S22 1.3100 0.2800 0.4490 0.0880 105.76 0.95 291.70 1.40 95.00 20.00 1996.900 10.200 540.0 63.0 16.60 5903.7 9.7 0.32 0.10
S23 0.2530 0.0120 0.5600 0.1400 48.00 7.10 249.00 13.00 39.00 6.70 2024.700 3.700 45.8 1.6 17.80 910.5 1.6 0.85 0.06
S24 0.9440 0.0480 0.8970 0.0049 103.67 0.42 7.93 0.37 290.00 15.00 2024.500 0.030 331.0 16.0 15.60 795.3 30.8 0.99 0.07
S29 0.4280 0.0190 0.7280 0.0520 105.80 1.70 161.96 0.80 346.50 5.90 2025.960 0.940 101.0 2.0 16.70 952.2 67.4 0.87 0.05
S31 0.4490 0.0100 0.5497 0.0025 109.03 0.27 137.16 0.30 308.00 3.00 2018.070 0.140 108.0 1.2 15.70 1653.7 14.6 0.63 0.02
S33 0.6570 0.0260 0.6080 0.0640 60.50 2.50 100.10 5.50 303.70 1.60 1928.000 12.000 192.0 5.2 16.00 2106.5 179.7 0.56 0.03
S38 0.1416 0.0002 0.8201 0.0007 171.10 2.10 101.06 0.24 17.99 0.25 2003.190 0.010 19.2 0.0 17.00 208.4 1.5 1.91 0.01
S39 0.3700 0.0150 0.9236 0.0021 89.36 0.73 159.03 0.10 23.30 3.80 2000.060 0.060 81.1 1.5 16.80 231.2 3.3 1.86 0.09
S42 0.9500 0.1800 0.5670 0.0830 67.16 0.66 196.14 0.75 35.80 3.20 2008.240 0.750 335.0 58.0 17.50 3364.4 24.8 0.44 0.13
S54 1.2000 0.8700 0.8930 0.0780 62.20 1.40 288.35 0.70 140.80 2.30 2004.460 0.070 477.0 199.0 17.50 1050.2 1.9 0.86 0.78
S55 0.1078 0.0010 0.7209 0.0077 150.10 2.20 325.50 4.00 331.50 3.90 2009.340 0.040 12.8 0.1 17.50 246.1 4.1 1.70 0.02
S60 0.3877 0.0070 0.7179 0.0051 126.87 0.30 170.54 0.85 29.37 0.29 2023.890 0.090 87.1 1.4 16.30 894.5 1.7 0.89 0.02
S66 1.5020 0.0950 0.1280 0.0430 128.50 1.60 92.30 3.20 134.00 17.00 1771.000 38.000 664.0 37.0 14.80 10712.4 620.5 0.21 0.02
S67 1.1260 0.0260 0.2930 0.0570 136.00 1.10 96.50 6.40 213.50 1.60 1705.000 22.000 431.0 10.0 12.10 6511.2 360.6 0.29 0.01
S71 0.9730 0.0400 0.8990 0.0130 74.00 1.30 35.16 0.86 337.80 4.90 1695.000 21.000 346.0 11.0 16.10 803.8 1.4 0.99 0.06
S83 1.4900 0.1900 0.3650 0.0750 127.20 1.40 87.70 1.20 203.60 6.00 2046.800 6.300 656.0 69.0 13.60 7738.6 22.5 0.27 0.05
S85 4.6000 3.3000 0.7800 0.1500 84.78 0.29 107.36 0.43 156.30 6.80 1930.200 9.800 3580.0 2550.0 15.60 8277.1 29.6 0.30 0.33
S87 2.7400 0.1600 0.2240 0.0270 119.54 0.87 106.32 0.99 336.10 7.70 611.000 154.000 1640.0 105.0 13.60 17390.5 2572.9 0.17 0.02
S89 1.0810 0.0550 0.6390 0.0380 87.61 0.16 238.99 0.18 126.40 4.00 1783.000 26.000 406.0 27.0 15.30 3191.8 407.2 0.46 0.04
S91 1.9170 0.0890 0.3030 0.0340 114.49 0.32 105.35 0.74 356.40 1.60 1108.000 69.000 958.0 50.0 12.20 10928.4 74.5 0.22 0.02
S96 1.4990 0.0570 0.1740 0.0220 126.36 0.96 115.66 0.59 233.60 2.40 1646.000 16.000 662.0 29.0 10.00 10127.0 530.0 0.22 0.02
S97 2.3200 0.4600 0.3500 0.1100 113.00 1.30 113.20 1.40 28.00 14.00 2132.000 29.000 1270.0 309.0 10.30 12333.9 305.9 0.21 0.08
S145 1.1200 0.1800 0.5000 0.2500 83.70 1.60 263.92 0.94 185.00 16.00 1808.000 58.000 426.0 71.0 17.50 4580.2 1471.2 0.37 0.10
S175 0.4140 0.0390 0.9867 0.0018 88.53 0.60 326.83 0.78 68.52 0.40 2009.510 0.010 96.2 5.0 17.50 45.0 0.8 4.27 0.47
R34 1.8100 0.1500 0.6410 0.0980 136.00 8.30 330.00 19.00 57.00 8.00 1522.000 52.000 877.0 83.0 14.00 5314.6 856.3 0.36 0.05
R44 3.9000 1.4000 0.2700 0.2700 131.00 5.20 80.50 7.10 217.00 24.00 1963.000 85.000 2730.0 1350.0 14.00 23285.6 901.5 0.15 0.11

Orbital elements from "An Update on Monitoring Stellar Orbits in the Galactic Center" (Gillesen et. al 2017) (arXiv:1611.09144), in particular this table from the supplementary data, with the exception of the parameters for the star S2 which were obtained from "A geometric distance measurement to the Galactic center black hole with 0.3% uncertainty" (GRAVITY 2019) (arXiv:1904.05721). The columns are:

id
The star's ID according to Gillesen
a, Δa
The semi-major axis in arcseconds, and its uncertainty.
e, Δe
The eccentricity and its uncertainty.
i, Δi
The inclination and its uncertainty, in degrees.
Ω, ΔΩ
The position angle of the ascending node and its uncertainty, in degrees.
ω, Δω
The longitude of pericenter and its uncertainty, in degrees.
Tp, ΔTp
The epoch of pericenter passage and its uncertainty, in fractional years.
P, ΔP
The orbital period and its uncertainty, in years.
Kmag
The star's K-band magnitude.
r, Δr
The pericenter distance and its uncertainty in AU.
v, Δv
The pericenter speed and its uncertainty, in percent of the speed of light.

The last 4 columns were computed based on the galactic center distance from (GRAVITY 2019) of R = 8179±13 pc. Uncertanties are only approximate as the joint distribution of the parameters was not available. Instead I expanded the expressions to first order in the uncertainty while ignoring correlations. The large effect of small changes in eccentricity as it approaches 1 would have been a problem for this, but this was mitigated by being provided with direct uncertainties in the angular pericenter distance (ρ=a(1-e)) (Gillessen, private communication) (see the last column in this table). With this, the expressions for the pericenter distance and speed are:

\begin{align} r &= R\rho = Ra(1-e) \\ \Delta r &\approx R\rho\sqrt{\frac{\Delta R^2}{R^2} + \frac{\Delta\rho^2}{\rho^2}} \\ v &= \frac{2\pi Ra}{P} \sqrt{\frac{2a}{\rho}-1} \\ \Delta v &\approx \frac{2\pi Ra}{P} \sqrt{ \Big(\frac{2a}{\rho}-1\Big)\Big(\frac{\Delta a^2}{a^2} + \frac{\Delta R^2}{R^2} + \frac{\Delta P^2}{P^2}\Big) + \frac{\Delta\rho^2}{4\rho^2} + \frac{\Delta a^2}{a^2}\Big(\frac14+\sqrt{\frac{2a}{\rho}-1}\Big) } \end{align}