Crab pulsar: IXPE observations reveal unified polarization properties in the optical and soft X-ray bands

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Table 1.

Summary of the model parameters we fit to three separate observations of the Crab pulsar by IXPE.

Obs.	α/Δt	β/α	b_I/Δt	b_Q/Δt	b_U/Δt	δ_I	δ_P	χ²/d.o.f. ^(a)
			[×10⁻² cnt/s]	[×10⁻² cnt/s]	[×10⁻² cnt/s]	[%]	[%]
1^st	0.930 ± 0.004	$0 . 55_{- 0.13}^{+ 0.12}$ $0.55^{+0.12}_{-0.13}$	72.00 ± 0.09	$0 . 62_{- 0.41}^{+ 0.42}$ $0.62^{+0.42}_{-0.41}$	$- 16 . 54_{- 0.41}^{+ 0.41}$ $-16.54^{+0.41}_{-0.41}$	$7 . 42_{- 4.42}^{+ 5.94}$ $7.42^{+5.94}_{-4.42}$	$- 0 . 67_{- 0.63}^{+ 0.51}$ $-0.67^{+0.51}_{-0.63}$	126.46/75
2^nd	0.957 ± 0.003	$0 . 53_{- 0.10}^{+ 0.10}$ $0.53^{+0.10}_{-0.10}$	76.65 ± 0.08	$- 0 . 55_{- 0.32}^{+ 0.32}$ $-0.55^{+0.32}_{-0.32}$	$- 17 . 40_{- 0.34}^{+ 0.34}$ $-17.40^{+0.34}_{-0.34}$	$21 . 23_{- 2.74}^{+ 3.52}$ $21.23^{+3.52}_{-2.74}$	$- 0 . 90_{- 0.70}^{+ 0.57}$ $-0.90^{+0.57}_{-0.70}$	95.65/75
3^rd	0.913 ± 0.007	$0 . 31_{- 0.21}^{+ 0.19}$ $0.31^{+0.19}_{-0.21}$	76.93 ± 0.13	$0 . 24_{- 0.56}^{+ 0.56}$ $0.24^{+0.56}_{-0.56}$	$- 16 . 98_{- 0.55}^{+ 0.54}$ $-16.98^{+0.54}_{-0.55}$	$- 2 . 16_{- 13.88}^{+ 9.13}$ $-2.16^{+9.13}_{-13.88}$	$- 1 . 13_{- 2.24}^{+ 1.61}$ $-1.13^{+1.61}_{-2.24}$	100.75/75

Model without phase shift

1^st	0.930 ± 0.004	$0 . 51_{- 0.12}^{+ 0.12}$ $0.51^{+0.12}_{-0.12}$	72.00 ± 0.09	$0 . 68_{- 0.41}^{+ 0.41}$ $0.68^{+0.41}_{-0.41}$	$- 16 . 66_{- 0.42}^{+ 0.40}$ $-16.66^{+0.40}_{-0.42}$	0	0	132.05/77
2^nd	0.957 ± 0.003	$0 . 42_{- 0.10}^{+ 0.10}$ $0.42^{+0.10}_{-0.10}$	76.65 ± 0.08	$- 0 . 48_{- 0.34}^{+ 0.33}$ $-0.48^{+0.33}_{-0.34}$	$- 17 . 70_{- 0.33}^{+ 0.35}$ $-17.70^{+0.35}_{-0.33}$	0	0	96.71/77
3^rd	0.913 ± 0.007	$0 . 44_{- 0.16}^{+ 0.16}$ $0.44^{+0.16}_{-0.16}$	76.93 ± 0.13	$0 . 49_{- 0.52}^{+ 0.55}$ $0.49^{+0.55}_{-0.52}$	$- 16 . 82_{- 0.52}^{+ 0.52}$ $-16.82^{+0.52}_{-0.52}$	0	0	101.07/77

Model including 90% subtraction of DC component from the Stokes parameters in the optical band

1^st	0.930 ± 0.004	$0 . 61_{- 0.14}^{+ 0.14}$ $0.61^{+0.14}_{-0.14}$	72.00 ± 0.09	$0 . 47_{- 0.41}^{+ 0.40}$ $0.47^{+0.40}_{-0.41}$	$- 17 . 11_{- 0.37}^{+ 0.37}$ $-17.11^{+0.37}_{-0.37}$	$2 . 91_{- 2.22}^{+ 1.36}$ $2.91^{+1.36}_{-2.22}$	$- 0 . 54_{- 0.56}^{+ 0.45}$ $-0.54^{+0.45}_{-0.56}$	123.59/75
2^nd	0.957 ± 0.003	$0 . 53_{- 0.10}^{+ 0.10}$ $0.53^{+0.10}_{-0.10}$	76.65 ± 0.08	$- 0 . 98_{- 0.31}^{+ 0.29}$ $-0.98^{+0.29}_{-0.31}$	$- 17 . 93_{- 0.31}^{+ 0.30}$ $-17.93^{+0.30}_{-0.31}$	$32 . 91_{- 1.51}^{+ 1.23}$ $32.91^{+1.23}_{-1.51}$	$- 0 . 99_{- 0.65}^{+ 0.57}$ $-0.99^{+0.57}_{-0.65}$	84.33/75
3^rd	0.913 ± 0.007	$0 . 54_{- 0.20}^{+ 0.18}$ $0.54^{+0.18}_{-0.20}$	76.93 ± 0.13	$0 . 42_{- 0.54}^{+ 0.53}$ $0.42^{+0.53}_{-0.54}$	$- 17 . 23_{- 0.48}^{+ 0.47}$ $-17.23^{+0.47}_{-0.48}$	$- 4 . 51_{- 7.17}^{+ 4.41}$ $-4.51^{+4.41}_{-7.17}$	$- 0 . 78_{- 1.09}^{+ 0.85}$ $-0.78^{+0.85}_{-1.09}$	95.96/75

Notes.

^(a)

χ² statistic reported for the set of parameters {β, b_Q, b_U, δ_I, δ_P}. The parameters {α, b_I} are fitted separately to the IXPE Stokes I; they do not depend on the treatment of the polarized components of the Stokes vector. The quantity Δt is the exposure time divided by the number of phase bins (40).

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