1111
1212
1313def _tau_of_u_exact_elliptic (
14- ecc : float , u : "npt.NDArray[np.float64]"
14+ ecc : float ,
15+ u : "npt.NDArray[np.float64]" ,
1516) -> "npt.NDArray[np.float64]" :
1617 cosqr , eusqrt = 1 - ecc ** 2 , np .sqrt (ecc ** 2 - u ** 2 )
1718 trig_numer = np .pi / 2 - np .arctan ((ecc ** 2 + u ) / np .sqrt (cosqr ) / eusqrt )
1819 return - eusqrt / cosqr / (1 + u ) + trig_numer / cosqr ** 1.5
1920
2021
2122def _tau_of_e_plus_u_elliptic (
22- ecc : float , u : "npt.NDArray[np.float64]"
23+ ecc : float ,
24+ u : "npt.NDArray[np.float64]" ,
2325) -> "npt.NDArray[np.float64]" :
2426 epu = np .sqrt (2 * (ecc + u ) / ecc )
2527 const = np .pi / (1 - ecc ** 2 ) ** 1.5
2628 return const - epu / (ecc - 1 ) ** 2 + epu ** 3 * (1 - 9 * ecc ) / 24 / (1 - ecc ) ** 3
2729
2830
2931def _tau_of_e_minus_u_elliptic (
30- ecc : float , u : "npt.NDArray[np.float64]"
32+ ecc : float ,
33+ u : "npt.NDArray[np.float64]" ,
3134) -> "npt.NDArray[np.float64]" :
3235 emu = np .sqrt (2 * (ecc - u ) / ecc )
3336 return emu / (1 + ecc ) ** 2 - emu ** 3 * (1 + 9 * ecc ) / 24 / (1 + ecc ) ** 3
@@ -49,7 +52,8 @@ def tau_of_u_parabolic(ecc: float, u: "npt.ArrayLike") -> "npt.NDArray[np.float6
4952
5053
5154def _tau_of_u_exact_hyperbolic (
52- ecc : float , u : "npt.ArrayLike"
55+ ecc : float ,
56+ u : "npt.ArrayLike" ,
5357) -> "npt.NDArray[np.float64]" :
5458 u = np .array (u , copy = False )
5559 cosqr , eusqrt = ecc ** 2 - 1 , np .sqrt (ecc ** 2 - u ** 2 )
@@ -58,7 +62,8 @@ def _tau_of_u_exact_hyperbolic(
5862
5963
6064def _tau_of_1_plus_u_hyperbolic (
61- ecc : float , u : "npt.NDArray[np.float64]"
65+ ecc : float ,
66+ u : "npt.NDArray[np.float64]" ,
6267) -> "npt.NDArray[np.float64]" :
6368 cosqr = ecc ** 2 - 1
6469 up1 = ecc * (1 + u ) / 2 / cosqr
@@ -68,7 +73,8 @@ def _tau_of_1_plus_u_hyperbolic(
6873
6974
7075def _tau_of_e_minus_u_hyperbolic (
71- ecc : float , u : "npt.NDArray[np.float64]"
76+ ecc : float ,
77+ u : "npt.NDArray[np.float64]" ,
7278) -> "npt.NDArray[np.float64]" :
7379 emu = np .sqrt (2 * (ecc - u ) / ecc )
7480 return emu / (1 + ecc ) ** 2 + emu ** 3 * (1 + 9 * ecc ) / 24 / (1 + ecc ) ** 3
@@ -96,7 +102,8 @@ def tau_of_u_prime2(ecc: float, u: "npt.ArrayLike") -> "npt.NDArray[np.float64]"
96102
97103
98104def esolve_u_from_tau_parabolic (
99- ecc : float , tau : "npt.ArrayLike"
105+ ecc : float ,
106+ tau : "npt.ArrayLike" ,
100107) -> "npt.NDArray[np.float64]" :
101108 tau = np .array (tau , copy = False )
102109 tau_3 = 3 * tau
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