---
title: "Recompute an EMRI frequency evolution from a published flux table"
description: "Run one Python script that turns the Fujita–Tagoshi flux table into wave cycles and frequency against time, then test it with three deliberate changes."
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language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/extreme-mass-ratio-inspirals/recompute-a-frequency-evolution/"
last_modified: "2026-11-12"
---

[Extreme mass ratio inspirals, from orbits to forecasts](https://sandboxphysics.com/stories/extreme-mass-ratio-inspirals/)

# Recompute an EMRI frequency evolution from a published flux table

Sandbox Physics · 2026-11-12

This tutorial recomputes how the wave frequency of a circular EMRI rises with time, using a frozen copy of a published Teukolsky flux table instead of the quadrupole formula. Everything needed is in one folder: the inputs, a standard-library Python script, the expected output and checksums. The run takes under a second.

## Before you begin

Python 3 and the energy-balance chapter of this course. No packages beyond the standard library.

## What you will reproduce

Three circular, equatorial inspirals with $q=10^{-5}$ and $M=10^6\,M_\odot$: zero spin from $r=10$ to just outside the ISCO, spin 0.9 from $r=10$ to $r=6$, and spin $-0.9$ from $r=20$ to $r=10$. For each, the script reports the time and the number of wave cycles with the table flux and with the quadrupole flux, and a frequency track with nine points.

| Case | Cycles, table flux | Years, table flux | Cycles, quadrupole flux | Ratio |
| --- | --- | --- | --- | --- |
| $a=0$, $10\to6.003$ | $112\,981$ | $1.417$ | $112\,045$ | $0.9917$ |
| $a=0.9$, $10\to6$ | $260\,673$ | $3.157$ | $222\,719$ | $0.8544$ |
| $a=-0.9$, $20\to10$ | $971\,669$ | $31.47$ | $970\,777$ | $0.9991$ |

## Step 1: the frozen flux table

Fujita and Tagoshi’s Table VII gives the gravitational-wave luminosity, summed through $\ell=6$, for circular equatorial orbits at seven spins from $-0.9$ to $0.9$. The inputs file holds the rows $r=6$ to $r=20$, transcribed by hand from arXiv version 1, with the caption quoted.

The table does not state its normalisation in a form we could quote, so the reading “energy flux to infinity per $(\mu/M)^2$” is our inference. Two checks support it. The mode-by-mode Table III at $r=6$ sums to the Table VII entry within $5\times10^{-5}$. Far out, the table agrees with the Tagoshi–Sasaki post-Newtonian flux to infinity within $2\times10^{-4}$ at $r\ge50$.

Reading: Fujita & Tagoshi (2004) · Tagoshi & Sasaki (1994)

## Step 2: interpolate a smooth correction, not the flux

The flux falls by a factor of a thousand across the table, so the script interpolates the ratio of the table flux to the quadrupole flux instead. That ratio stays between 0.8 and 2 and changes smoothly with radius. The script fits a cubic through the four nearest tabulated radii and refuses any radius outside 6 to 20.

The energy slope $E'(r)$ comes from the exact Kerr circular energy by a fourth-order finite difference. Both integrals use Simpson’s rule in $\ln r$ with 2000 intervals.

$$
t=\int_{\ln r_1}^{\ln r_0}\frac{r\,E'(r)}{\mathcal F(r)}\,d\ln r,\qquad N=\int_{\ln r_1}^{\ln r_0}\frac{\Omega_\phi(r)}{\pi}\,\frac{r\,E'(r)}{\mathcal F(r)}\,d\ln r,\qquad \mathcal F=\tfrac{32}{5}\,q\,\Omega_\phi^{10/3}\times\frac{\mathcal F_{\rm table}}{\mathcal F_{\rm quad}}
$$

Integrating in $\ln r$ spreads the points evenly where the integrand changes most.

## Step 3: run and compare

Download the folder, check the checksums, run the script with the inputs file and a new output name, then compare your output with the expected one. On the machine that produced the expected file, Python 3.9.6 on macOS, a fresh run reproduces it byte for byte. Elsewhere, expect agreement to about one part in a trillion.

Time is converted to seconds with $GM_\odot/c^3=4.925490947\times10^{-6}\,\mathrm s$ and to years with the Julian year of 31,557,600 seconds. The wave frequency in the track is $f=2\Omega_\phi/(2\pi)$, in hertz for $M=10^6\,M_\odot$.

## Step 4: break it on purpose

The output contains three deliberate changes and how much each moves the zero-spin cycle count. Linear instead of cubic interpolation of the correction changes it by −0.54%. Four Simpson intervals instead of 2000 change it by +0.088%. Replacing the Kerr energy with the Newtonian binding energy changes it by +99%.

The last number is the lesson: near the ISCO the binding energy matters as much as the flux. A fair test of a flux model has to keep the orbit exact.

## What the result cannot tell you

Extrapolation is refused. Spin 0.9 has its ISCO at $2.32$, far below the table’s first row, so that case stops at $r=6$. The request for $r=4$ is listed in the output as refused.

Missing multipoles. The $\ell=6$ modes carry 0.29% of the zero-spin flux at $r=6$, so the omitted $\ell\ge7$ modes are likely about 0.1% there. That is an extrapolation of the trend, not a calculation.

Missing physics. The horizon flux, eccentric and inclined orbits, resonances and every post-adiabatic term are absent. The post-adiabatic phase correction is of order one radian, so the cycle counts are right to the leading order in the mass ratio only.

Reading: Fujita & Tagoshi (2004) · Barack & Pound (2019)

## Download and run

Put the six files in one folder, then run these two commands. The second writes your own output next to the expected one.

`shasum -a 256 -c SHA256SUMS python3 recompute_frequency_evolution.py inputs.json my-output.json`

- [recompute\_frequency\_evolution.py](https://sandboxphysics.com/stories/extreme-mass-ratio-inspirals/materials/frequency-evolution/recompute_frequency_evolution.py) · The script, Python standard library only
- [inputs.json](https://sandboxphysics.com/stories/extreme-mass-ratio-inspirals/materials/frequency-evolution/inputs.json) · Cases, units and the frozen flux table
- [expected-output.json](https://sandboxphysics.com/stories/extreme-mass-ratio-inspirals/materials/frequency-evolution/expected-output.json) · Expected output, Python 3.9.6
- [README.txt](https://sandboxphysics.com/stories/extreme-mass-ratio-inspirals/materials/frequency-evolution/README.txt) · Units, model and failure cases
- [SHA256SUMS](https://sandboxphysics.com/stories/extreme-mass-ratio-inspirals/materials/frequency-evolution/SHA256SUMS) · SHA-256 checksums of every file
- [LICENSE.txt](https://sandboxphysics.com/stories/extreme-mass-ratio-inspirals/materials/frequency-evolution/LICENSE.txt) · Permission and attribution

## Try it in the Lab

### [Compare with the Lab’s quadrupole count](https://sandboxphysics.com/labs/emri-spacetime-mapping/)

1. Choose zero spin, eccentricity zero and $p=10$ in the Lab.
2. Read the wave cycles left. The Lab rounds to three figures; compare it with 112,045 in the quadrupole column of the first row.

The Lab starts its count at the orbit size you set and uses the quadrupole flux only.

## Exercises and explained answers

Which case shows the largest gap between the table and the quadrupole flux, and why that one?

Spin 0.9 from $r=10$ to $r=6$, with a ratio of 0.854. On that range the quadrupole flux is 15% to 19% too large everywhere, so the error never changes sign and does not cancel. For zero spin the error changes sign near $r=8$.

Change the zero-spin stop radius in the inputs to 5.9 and rerun. What happens?

The script stops with an error that names the radius and refuses to extrapolate, because $r=5.9$ lies below the first tabulated row. Even with a larger table it would also lie inside the zero-spin ISCO at $r=6$, where no circular orbit is stable.

## Primary sources and what we read in each

1. [Barack & Pound (2019), Self-force and radiation reaction in general relativity, Rep. Prog. Phys., doi:10.1088/1361-6633/aae552](https://arxiv.org/abs/1805.10385v2)

   arXiv version 2, dated 19 November 2018. Section 5.2 and eqs. (72)–(74): the two-timescale phase expansion and what the adiabatic and post-adiabatic orders need; Section 5.3 on transient resonances; Section 6.1 and eqs. (77)–(78), the orbit-averaged balance laws, including negative horizon fluxes.

2. [Fujita & Tagoshi (2004), New numerical methods to evaluate homogeneous solutions of the Teukolsky equation, Prog. Theor. Phys. 112, 415](https://arxiv.org/abs/gr-qc/0410018v1)

   arXiv version 1. Table VII, the gravitational-wave luminosity through l = 6 for circular equatorial orbits at seven spins; Table III, the mode-by-mode luminosity at r = 6M. The unit normalisation and the reading as flux to infinity are our inference, checked against the quadrupole and post-Newtonian limits.

3. [Tagoshi & Sasaki (1994), Post-Newtonian expansion of gravitational waves from a particle in circular orbit around a Schwarzschild black hole, Prog. Theor. Phys. 92, 745](https://arxiv.org/abs/gr-qc/9405062v1)

   arXiv version 1. Eq. (43): the energy flux to infinity relative to the quadrupole value, through fourth post-Newtonian order; used only as an independent weak-field check of the flux table.

## Keep reading

Extreme mass ratio inspirals · Reading path

### A small black hole maps a large one

1. [Article **What hundreds of thousands of orbits can reveal about a black hole**](https://sandboxphysics.com/stories/extreme-mass-ratio-inspirals/what-hundreds-of-thousands-of-orbits-reveal/)
2. [Article **How orbits precess around a spinning black hole and where they stop being stable**](https://sandboxphysics.com/stories/extreme-mass-ratio-inspirals/geodesics-precession-and-the-last-stable-orbit/)
3. [Article **How energy loss sets the cycles left before the last stable orbit**](https://sandboxphysics.com/stories/extreme-mass-ratio-inspirals/radiation-reaction-and-the-adiabatic-inspiral/)
4. [Article **Why the wave from an eccentric orbit is a comb of lines**](https://sandboxphysics.com/stories/extreme-mass-ratio-inspirals/fundamental-frequencies-and-harmonics/)
5. [You are here **Recompute an EMRI frequency evolution from a published flux table**](https://sandboxphysics.com/stories/extreme-mass-ratio-inspirals/recompute-a-frequency-evolution/)
6. [Article **Reading the EMRI literature on forecasts, self-force, resonances and multipole tests**](https://sandboxphysics.com/stories/extreme-mass-ratio-inspirals/reading-emri-forecasts-self-force-resonances-multipoles/)