---
title: "Follow fuel across the disk boundary"
description: "Derive how a moving gas population changes disk star formation, while the global rate can remain identical across different wind paths."
canonical_url: "https://sandboxphysics.com/stories/ram-pressure-stripping/follow-fuel-across-the-disk-boundary/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/ram-pressure-stripping/follow-fuel-across-the-disk-boundary/"
last_modified: "2026-11-11"
---

[Ram pressure and returning gas](https://sandboxphysics.com/stories/ram-pressure-stripping/)

Ram pressure and returning gas · L2

# Follow fuel across the disk boundary

Sandbox Physics · 2026-11-11

The star-formation rate inside a disk can fall because gas leaves the region being counted. To decide whether the whole galaxy has stopped forming stars, follow the same gas outside that boundary and state what formation law it obeys.

## Before you begin

You need mass, rates and the distinction between position and binding. An exponential will appear as the solution of a constant fractional consumption rate.

## Start with an aperture, not a headline

An aperture is simply the region over which a measurement is added. A telescope aperture can be a shape on an image; a simulation can define a physical volume. Here the disk aperture is a cylinder of radius 16 kiloparsecs and half-height 1 kiloparsec, centred on the galaxy.

At each time, mark which gas sheets occupy that cylinder. Add their current masses. A sheet lifted just beyond the upper face no longer contributes to the disk total even if it is still gravitationally bound. When it returns, it contributes again.

The aperture is deliberately geometric. It does not select molecular gas, apply an emission threshold or decide which gas is actively forming stars. Those require additional physical prescriptions or observations. A useful comparison begins by naming exactly which selection has been made.

Reading: Original fixed-slice and static-restoring-force materials

## Give every sheet the same consumption rule

In this model, every gas sheet consumes a fixed fraction of its remaining mass per unit time. The chosen time scale is 2000 million years, and the returned mass fraction is zero. Material removed from the gas enters the accumulated birth-mass inventory.

If a sheet contains twice as much gas, it forms stars twice as fast. After one consumption time it retains the fraction one divided by the natural exponential base, about 36.8%. The rate slows as the reservoir becomes smaller.

This rule follows a sheet wherever it moves. The wind moves fuel but does not switch the formation rule off in the tail. Summing all sheets therefore gives exactly the same all-gas rate for every supported wind path.

$$
\begin{aligned}\dot m_i&=-\frac{m_i}{\tau_\star},\qquad\tau_\star=2000\,\mathrm{Myr}\\m_i(T)&=m_{i,0}\exp(-T/\tau_\star)\end{aligned}
$$

$T$ is elapsed time since the common initial state. Model time is $t=T-500\,\mathrm{Myr}$. Here $\dot m_i$ describes removal from gas, with the same amount added to birth mass.

Reading: Original fixed-slice and static-restoring-force materials

## Count the moving sheets at one common time

Introduce an indicator: it equals one when a sheet is inside the disk cylinder and zero otherwise. Multiply the sheet’s rate by that indicator before summing. A change in disk star formation can now come from two places: ordinary fuel consumption, or sheets crossing the aperture.

At model time 302.34375 million years, the weak $q=2.72$ passage has a disk rate of about 0.2142 solar masses per year. Its all-gas rate is 0.2176. For $q=4$, which has zero centre-sampled incident density throughout the encounter, both rates are 0.2176. The smaller disk value in $q=2.72$ comes from gas just outside the disk.

The all-gas equality is not a discovery about real galaxies. It follows from the formation rule we specified. It makes a controlled exercise: keep the global consumption identical and isolate the effect of where the fuel lies.

$$
{\rm SFR}_{\mathcal D}(t)=\sum_i\mathbf1_{\mathcal D}(\mathbf x_i(t))\frac{m_i(t)}{\tau_\star}
$$

$\mathbf1_{\mathcal D}$ is the disk indicator. Convert the final rate from solar masses per million years to solar masses per year by dividing by one million.

Original synthetic slices at the same model time $302.34375\,\mathrm{Myr}$. All-gas formation agrees because each sheet obeys the same specified consumption law. The disk count differs because the chosen cylinder excludes some bound gas. Open the full figure

Reading: Original fixed-slice and static-restoring-force materials

## A ninety-percent radius is not a sharp edge

Gas need not remain in one continuous annulus. Some radial sheets can be displaced while others at larger cylindrical radii are present inside the volume. Calling the outermost remaining sample “the stripping radius” would hide these gaps.

Instead, sort the currently resident sheets by their current cylindrical radius, add their current gas masses, and stop where the accumulated mass reaches 90% of the resident total. That defines the disk-resident ninety-percent mass radius.

The statistic describes a mass distribution inside a chosen volume. It can grow while some gas is outside, and it can change when returning gas crosses the boundary. Neither its movement nor its numerical agreement with a static reference automatically identifies an escape boundary.

Reading: Original fixed-slice and static-restoring-force materials

## Ask which extra physics would change the answer

A real tail may contain gas too hot or too diffuse to form stars at the disk rate. Compression, molecular chemistry, cooling, mixing and feedback can all change the relation between gas mass and newly formed stars. Measuring a global shutdown requires a model or data that follows those changes.

For this exercise, add the disk and outside formation rates and check that their sum equals the all-gas rate. Then imagine setting the outside rate to zero. That would be a different physical prescription, requiring a reason and a new calculation, not a new label on the same curve.

This separation gives a practical habit for observations as well: note the spatial selection, identify the gas or light tracer, then compare like with like. Fuel transport and a change in the efficiency of making stars can act together, but they enter the explanation at different steps.

Reading: Original fixed-slice and static-restoring-force materials · Gronke & Oh, The growth and entrainment of cold gas in a hot wind, arXiv:1806.02728v2 (2018)

## Exercises and explained answers

1. The disk rate rises while every individual sheet consumes gas. How can both be true?

   Read the explanation

   Previously displaced gas can re-enter the aperture. The larger amount included in the disk sum can outweigh the gradual decline of each sheet’s mass.

2. Does a larger $R_{90}$ always mean more gas has escaped?

   Read the explanation

   No. $R_{90}$ uses only current disk-resident gas and its current radii. Escape requires a separate energy/history test.

## Try the prediction

### [Follow gas lifted by a cluster wind](https://sandboxphysics.com/labs/ram-pressure-stripping/)

1. At the same native time, compare $q=2.72$ with the zero-centre-wind path and explain the two SFR values.
2. Follow a returning sheet through the cylinder boundary and predict the direction of the disk-rate change.

[Previous: From a wind pulse to a restoring force](https://sandboxphysics.com/stories/ram-pressure-stripping/from-a-wind-pulse-to-restoring-force/) [Next: Recompute five fixed gas slices](https://sandboxphysics.com/stories/ram-pressure-stripping/recompute-five-fixed-gas-slices/)

## Sources and calculation scope

1. [Gronke & Oh, The growth and entrainment of cold gas in a hot wind, arXiv:1806.02728v2 (2018)](https://arxiv.org/abs/1806.02728v2)

   Controlled three-dimensional cloud-in-wind cooling study. Mixed-gas cooling, material origin, downstream domain and resolution are the relevant reading questions. It is not a self-gravitating star-forming disk or a calibration of this gas-sheet model.

2. [Original fixed-slice and static-restoring-force materials](https://sandboxphysics.com/stories/ram-pressure-stripping/materials/ram-pressure-science-v1.zip)

   Our five exact synthetic slices with sixty-four sheets and thirteen fields, initial common arrays, two global-light histories/three screens/four-age band subset, independent Python static maxima and mask/energy/colour diagnostics. Original trajectory and native SSP production are distinct earlier calculations; this offline package repeats diagnostics and band ratios. No browser Engine, original FITS/SED, paper, author code or observed-galaxy data is included.