Explanation of the exceptionally strong timing noise of PSR J0337+1715 by a circum-ternary planet and consequences for gravity tests

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Table E.1.

Mean values of the MCMC runs of the PL3 and Planet models with their 68% median confidence intervals.

Parameter	Symbol	PL3GR	PlanetGR
Fixed values

Reference epoch (MJD)	T_ref	56492
Position epoch (MJD)	T_pos	57205

Fitted values

Right ascension	α	$3^{h} 37^{m} 43^{s} . 82700 {(43)}_{- 67}^{+ 66}$ $3^{\mathrm{h}}37^{\mathrm{m}}43^{\mathrm{s}}.82700(43)_{-67}^{+66}$	$3^{h} 37^{m} 43^{s} . 82700 {(94)}_{- 64}^{+ 63}$ $3^{\mathrm{h}}37^{\mathrm{m}}43^{\mathrm{s}}.82700(94)_{-64}^{+63}$
Declination	δ	$17^{\deg} 15^{m} 14^{s} . 817 {(63)}_{- 37}^{+ 38}$ $17^{\mathrm{deg}}15^{\mathrm{m}}14^{\mathrm{s}}.817(63)_{-37}^{+38}$	$17^{\deg} 15^{m} 14^{s} . 817 {(75)}_{- 33}^{+ 40}$ $17^{\mathrm{deg}}15^{\mathrm{m}}14^{\mathrm{s}}.817(75)_{-33}^{+40}$
Distance (kpc)	d	$1160_{- 51}^{+ 53}$ $1160_{-51}^{+53}$	$1182_{- 55}^{+ 53}$ $1182_{-55}^{+53}$
Right-ascension proper motion (mas yr⁻¹)	μ_α	$6.1 {(37)}_{- 68}^{+ 70}$ $6.1(37)_{-68}^{+70}$	$5.9 {(82)}_{- 80}^{+ 70}$ $5.9(82)_{-80}^{+70}$
Declination proper motion (mas yr⁻¹)	μ_δ	$- 5 . {(34)}_{- 26}^{+ 25}$ $-5.(34)_{-26}^{+25}$	$- 4 . {(83)}_{- 25}^{+ 29}$ $-4.(83)_{-25}^{+29}$
Radial proper motion (mas yr⁻¹)	μ_d	$5 . {(63)}_{- 32}^{+ 31}$ $5.(63)_{-32}^{+31}$	$5 . {(52)}_{- 28}^{+ 32}$ $5.(52)_{-28}^{+32}$
Dispersion measure (pc cm⁻³)	DM	$21.316 {(46)}_{- 12}^{+ 12}$ $21.316(46)_{-12}^{+12}$	$21.316 {(46)}_{- 11}^{+ 11}$ $21.316(46)_{-11}^{+11}$
Dispersion measure variation (pc cm⁻³ yr⁻¹)	DM′	${(1.7)}_{- 2.1}^{+ 2.1} \times 10^{- 5}$ $(1.7)_{-2.1}^{+2.1} \times 10^{-5}$	${(1.6)}_{- 2.0}^{+ 1.9} \times 10^{- 5}$ $(1.6)_{-2.0}^{+1.9} \times 10^{-5}$
Re-scaled spin frequency (Hz)	$\bar{f}$ $\bar{f}$	$365.953337902243 {(85)}_{- 81}^{+ 81}$ $365.953337902243(85)_{-81}^{+81}$	$365.9533379022 {(56)}_{- 11}^{+ 11}$ $365.9533379022(56)_{-11}^{+11}$
Re-scaled spin frequency derivative (10⁻¹⁵ Hz s⁻¹)	$\bar{f}'$ $\bar{f}{\prime}$	$- 2.35631 {(40)}_{- 45}^{+ 48} \times 10^{- 15}$ $-2.35631(40)_{-45}^{+48} \times 10^{-15}$	$- 2.356 {(45)}_{- 12}^{+ 11} \times 10^{- 15}$ $-2.356(45)_{-12}^{+11} \times 10^{-15}$

Inner orbit I of pulsar p w.r.t. inner-binary centre of mass b

Orbital period (d)	P_I	$1.62939900 {(75)}_{- 42}^{+ 42}$ $1.62939900(75)_{-42}^{+42}$	$1.62939900 {(83)}_{- 41}^{+ 38}$ $1.62939900(83)_{-41}^{+38}$
Projected semi-major axis (lt-s)	a_psini_I	$1.2175276 {(17)}_{- 56}^{+ 56}$ $1.2175276(17)_{-56}^{+56}$	$1.2175276 {(25)}_{- 56}^{+ 54}$ $1.2175276(25)_{-56}^{+54}$
Inclination offset (°)	δi = i_I − i_O	$1 . 2_{- 2.4}^{+ 2.3} \times 10^{- 3}$ $1.2_{-2.4}^{+2.3} \times 10^{-3}$	$8_{- 23}^{+ 21} \times 10^{- 4}$ $8_{-23}^{+21} \times 10^{-4}$
Laplace-Lagrange	e_Icosω_p	$6.937 {(38)}_{- 40}^{+ 40} \times 10^{- 4}$ $6.937(38)_{-40}^{+40} \times 10^{-4}$	$6.937 {(32)}_{- 42}^{+ 37} \times 10^{- 4}$ $6.937(32)_{-42}^{+37} \times 10^{-4}$
Laplace-Lagrange	e_Isinω_p	$- 8.59 {(37)}_{- 41}^{+ 40} \times 10^{- 5}$ $-8.59(37)_{-41}^{+40} \times 10^{-5}$	$- 8.59 {(42)}_{- 35}^{+ 36} \times 10^{- 5}$ $-8.59(42)_{-35}^{+36} \times 10^{-5}$
Time of ascending node (MJD)	t_asc_p	$55917.15894 {(93)}_{- 19}^{+ 19}$ $55917.15894(93)_{-19}^{+19}$	$55917.15894 {(89)}_{- 17}^{+ 19}$ $55917.15894(89)_{-17}^{+19}$
Long. of asc. nodes offset (°)	δΩ = Ω_b − Ω_p	${(1.7)}_{- 1.3}^{+ 1.3} \times 10^{- 4}$ $(1.7)_{-1.3}^{+1.3} \times 10^{-4}$	${(1.5)}_{- 1.2}^{+ 1.2} \times 10^{- 4}$ $(1.5)_{-1.2}^{+1.2} \times 10^{-4}$

Outer orbit O of b w.r.t. triple-system centre of mass t

Orbital period (d)	P_O	$327.25512 {(55)}_{- 10}^{+ 10}$ $327.25512(55)_{-10}^{+10}$	$327.25512 {(61)}_{- 11}^{+ 10}$ $327.25512(61)_{-11}^{+10}$
Projected semi-major axis (lt-s)	a_bsini_O	$74.672344 {(35)}_{- 23}^{+ 22}$ $74.672344(35)_{-23}^{+22}$	$74.672344 {(31)}_{- 22}^{+ 21}$ $74.672344(31)_{-22}^{+21}$
Co-projected semi-major axis (lt-s)	a_bcosi_O	$91.4 {(10)}_{- 37}^{+ 37}$ $91.4(10)_{-37}^{+37}$	$91.4 {(02)}_{- 32}^{+ 35}$ $91.4(02)_{-32}^{+35}$
Laplace-Lagrange	e_Ocosω_b	$0.03511469 {(40)}_{- 27}^{+ 28}$ $0.03511469(40)_{-27}^{+28}$	$0.03511469 {(43)}_{- 27}^{+ 28}$ $0.03511469(43)_{-27}^{+28}$
Laplace-Lagrange	e_Osinω_b	$- 0.00352492 {(40)}_{- 18}^{+ 19}$ $-0.00352492(40)_{-18}^{+19}$	$- 0.00352492 {(22)}_{- 18}^{+ 19}$ $-0.00352492(22)_{-18}^{+19}$
Time of ascending node (MJD)	t_asc_b	$56230.195329 {(47)}_{- 82}^{+ 83}$ $56230.195329(47)_{-82}^{+83}$	$56230.195329 {(01)}_{- 78}^{+ 79}$ $56230.195329(01)_{-78}^{+79}$
Longitude of ascending node (°)	Ω_b	$145 . 9_{- 1.5}^{+ 1.4}$ $145.9_{-1.5}^{+1.4}$	$148 . 0_{- 1.5}^{+ 1.7}$ $148.0_{-1.5}^{+1.7}$

Inner mass ratio	m_i/m_p	$0.1373 {(76)}_{- 12}^{+ 12}$ $0.1373(76)_{-12}^{+12}$	$0.1373 {(77)}_{- 12}^{+ 12}$ $0.1373(77)_{-12}^{+12}$
ToA uncertainty re-scaling	EFAC	$1.11 {(77)}_{- 73}^{+ 76}$ $1.11(77)_{-73}^{+76}$	$1.11 {(72)}_{- 79}^{+ 75}$ $1.11(72)_{-79}^{+75}$

Derived values

Parallel proper motion (km s⁻¹)	V_∥	$29 . 77_{- 1.1}^{+ 0.90}$ $29.77_{-1.1}^{+0.90}$	$29 . {(72)}_{- 83}^{+ 76}$ $29.(72)_{-83}^{+76}$
Plane-of-sky proper motion (km s⁻¹)	V_⊥	$43 . 0_{- 2.0}^{+ 2.2}$ $43.0_{-2.0}^{+2.2}$	$41 . 4_{- 2.2}^{+ 2.1}$ $41.4_{-2.2}^{+2.1}$

Spin frequency (Hz)	f	$365.9533629 {(82)}_{- 13}^{+ 13}$ $365.9533629(82)_{-13}^{+13}$	$365.9533629 {(80)}_{- 12}^{+ 12}$ $365.9533629(80)_{-12}^{+12}$
Spin frequency derivative (10⁻¹⁵ Hz s⁻¹)	$\dot{f}$ $\dot f$	$- 2.29 {(06)}_{- 40}^{+ 43} \times 10^{- 15}$ $-2.29(06)_{-40}^{+43} \times 10^{-15}$	$- 2.29 {(67)}_{- 43}^{+ 41} \times 10^{- 15}$ $-2.29(67)_{-43}^{+41} \times 10^{-15}$

Inner orbit I of p/b

Semi-major axis (lt-s)	a_p	$1.924 {(46)}_{- 48}^{+ 48}$ $1.924(46)_{-48}^{+48}$	$1.924 {(39)}_{- 43}^{+ 46}$ $1.924(39)_{-43}^{+46}$
Orbital inclination (°)	i_I	$39.2 {(47)}_{- 11}^{+ 11}$ $39.2(47)_{-11}^{+11}$	$39.2 {(48)}_{- 11}^{+ 10}$ $39.2(48)_{-11}^{+10}$
Orbital eccentricity	e_I	$6.990 {(40)}_{- 40}^{+ 40} \times 10^{- 4}$ $6.990(40)_{-40}^{+40} \times 10^{-4}$	$6.990 {(36)}_{- 42}^{+ 37} \times 10^{- 4}$ $6.990(36)_{-42}^{+37} \times 10^{-4}$
Longitude of periastron (°)	ω_p	$- 7.06 {(16)}_{- 33}^{+ 33}$ $-7.06(16)_{-33}^{+33}$	$- 7.06 {(20)}_{- 28}^{+ 29}$ $-7.06(20)_{-28}^{+29}$
Time of periastron passage (MJD)	t_p_I	$55917.1269 {(88)}_{- 15}^{+ 15}$ $55917.1269(88)_{-15}^{+15}$	$55917.1269 {(86)}_{- 13}^{+ 13}$ $55917.1269(86)_{-13}^{+13}$
Longitude of asc. node (°)	Ω_p	$145 . 9_{- 1.5}^{+ 1.4}$ $145.9_{-1.5}^{+1.4}$	$148 . 0_{- 1.5}^{+ 1.7}$ $148.0_{-1.5}^{+1.7}$

Outer orbit O of b w.r.t. triple-system centre of mass t
Semi-major axis (lt-s)	a_b	$1.180 {(33)}_{- 29}^{+ 28} \times 10^{2}$ $1.180(33)_{-29}^{+28} \times 10^{2}$	$1.180 {(27)}_{- 25}^{+ 27} \times 10^{2}$ $1.180(27)_{-25}^{+27} \times 10^{2}$
Orbital inclination (°)	i_O	$39.2 {(45)}_{- 11}^{+ 11}$ $39.2(45)_{-11}^{+11}$	$39.2 {(48)}_{- 10}^{+ 10}$ $39.2(48)_{-10}^{+10}$
Orbital eccentricity	e_O	$0.03529117 {(20)}_{- 29}^{+ 29}$ $0.03529117(20)_{-29}^{+29}$	$0.03529117 {(21)}_{- 28}^{+ 29}$ $0.03529117(21)_{-28}^{+29}$
Longitude of periastron (°)	ω_b	$- 5.73232 {(85)}_{- 27}^{+ 27}$ $-5.73232(85)_{-27}^{+27}$	$- 5.73232 {(56)}_{- 27}^{+ 27}$ $-5.73232(56)_{-27}^{+27}$
Time of periastron passage (MJD)	t_p_O	$56224.98440 {(20)}_{- 19}^{+ 19}$ $56224.98440(20)_{-19}^{+19}$	$56224.98440 {(42)}_{- 19}^{+ 19}$ $56224.98440(42)_{-19}^{+19}$

Pulsar mass (M_⊙)	m_p	$1.43 {(82)}_{- 12}^{+ 12}$ $1.43(82)_{-12}^{+12}$	$1.43 {(80)}_{- 11}^{+ 12}$ $1.43(80)_{-11}^{+12}$
Inner-companion mass (M_⊙)	m_i	$0.197 {(58)}_{- 15}^{+ 15}$ $0.197(58)_{-15}^{+15}$	$0.197 {(55)}_{- 14}^{+ 15}$ $0.197(55)_{-14}^{+15}$
Outer-companion mass (M_⊙)	m_o	$0.410 {(18)}_{- 31}^{+ 31}$ $0.410(18)_{-31}^{+31}$	$0.410 {(12)}_{- 29}^{+ 31}$ $0.410(12)_{-29}^{+31}$

Error bars apply to the digits between parenthesis and delimit the 68% median confidence region, namely the interval excluding 16% of the distribution above and below it. The central value quoted is the best-fit value. Upper-case indices I, O refer to the inner and outer binaries respectively. Lower-case indices p, i, o refer to the pulsar, inner white dwarf, and outer white dwarf, respectively; b refers to the inner-binary centre of mass. Data spans MJD 56492 − 59480. Solar System ephemeris is DE430. Model-specific parameters are reported separately in Table 1.

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