ΔT is the difference between the terrestrial time (TT) and universal time (UT1). See my dynamical time page for a brief explanation of these terms.
This package provides a python code to implement the ΔT calculation according to the fitting and extrapolation formulae in Stephenson et al (2016) and Morrison et al (2021). Specifically, values of ΔT from -720 to 2022 are computed using their spline fit cubic polynomials (archived), but the coefficients after 2013 have been modified to extend the fitting formulae to 2022 (their original formulae are fitted up to 2019). Outside this range ΔT is extrapolated by integrating their long-term lod function. The uncertainty in ΔT is estimated based on their tables (archived) for years between -2000 to +2500. Outside this range, the error is estimated using quadratic functions. However, they are probably not reliable.
The cubic spline polynomials are fitted to the ΔT data from -720 to 2022. Daily values of ΔT in 2022 and later can be constructed from the data in IERS bulletins, which can be used to calculate the error of the extrapolation formula after 2022 and to extend the fitting formula. The csv file DeltaT_IERS.csv contains the ΔT data from 1972 to the recent time computed from the IERS data.
All functions that compute ΔT are in DeltaT.py and DeltaT_hybrid.py. DeltaT(y) computes ΔT (in seconds) as a function of year y using the fitting and extrapolation formulae. DeltaT_error_estimate(y) provides an error estimate of ΔT. DeltaT_with_error_estimate(y) combines the two functions and returns ΔT(y) with an error estimate. DeltaT_hybrid(jd) calculates ΔT by linear interpolation from the IERS data if the Julian date jd is in inside the range of the IERS data. Outside the data range, the function computes ΔT by the cubic spline polynomials or by the integrated long-term lod function. DeltaT_hybrid_error_estimate(jd) estimates the error of the computed ΔT. Inside the IERS data range, the error is estimated by either the measurement error of UT1 or half of the difference between two values of ΔT used in the interpolation, whichever is larger. The input y and jd can be a scalar or a 1D array.
The functions in regression_cubic_spline.py can be used to update the current fitting formulae.
The notebook DeltaT_examples.ipynb shows examples of using these functions.
A javascript version of the code is used in my eclipse website.
-
F.R. Stephenson, L.V. Morrison, and C.Y. Hohenkerk, Measurement of the Earth's rotation: 720 BC to AD 2015, Proc. R. Soc. A., 472:20160404 (2016).
-
L.V. Morrison, F.R. Stephenson, C.Y. Hohenkerk, and M. Zawilski, Addendum 2020 to 'Measurement of the Earth's rotation: 720 BC to AD 2015', Proc. R. Soc. A., 477:20200776 (2021).