Introduction

Frequency_4 is a collection of codes to compute frequency and time transfer in the Solar System with the precision up to 4-th order of $v/c$. It has been developed as a part of RSCF grant project "Exploring second-order gravitational effects and the possibility for the dark matter detection with the next generation of space-borne atomic standards of time and frequency". Codes provided in Fortran-90 and Python 3.x (requires Numpy). They represent simple but quite long algebraic expressions describing deflection of light, change of frequency and time for a photon emitted at point A and received at point B. We use Parametrized Post Newtonian (PPN) metrics for all the formulas.

Definitions

The gravitating body is located at the origin of coordinate system. Point of photon emission A has coordinates X0, point where it is received, B, has coordinates X1. The axis of symmetry of the gravitating body and the axis of rotation coincide with $z$ coordinate axis. The metric is characterized by 5 PPN parameters: $\gamma$, $\beta$, $\xi$, $\alpha_1$, $\alpha_2$.

Usage

Light deflection

Light deflection vector $\alpha_{x,y,z}$ is computed for a number of components. It is up to user to multiply them by the amplitudes, and take into account unints of measure. An example usage is given in example.f90 file.

For every component the deflection components are given by 3 functions

alpha_{i}_{c}(X0, X1)

where {i} is x,y or z, X0, X1 are the 3-element arrays of coordinates, and {c} is one of:

• 'r1' for 1/r potentian,
• 'r2' for 1/r^2 potential,
• 'r4' for 1/r^4 potential,
• 'quadrupole' for the quadrupole potential,
• 'V' for the vector potential $V_j$.

Also for the terms containing velocity relative to the CMB reference frame $w^j$:

alpha_{i}_{c}(X0, X1, wvec)

where wvec is the 3-component vector of $w^j$, and possible values of {c} are:

• 'wU' for the terms, proportional to $\alpha_1$ PPN coeffitient,
• 'wV' for the term, containing $\alpha_1 w^k V_k$,
• 'wUjk' for the terms, containing tensor potential $U_{jk}$ and proportional to the $\alpha_2$ coefficient

For example, to compute the actual deflection in x axis due to quadrupole only for the body of mass $M$, equatorial radius $R$ and dimensionless coefficient of second zonal harmonic $J_2$ the code is:

2*(1+ppn_gamma)*M*J_2*R**2 * alpha_i_quadrupole(X0,X1)

Compilation

The example.f90 can be compiled with the command:

gfortran freq_4.f90 example.f90 -o example.f90