---
title: "Which disk surface a light path meets first"
description: "Trace a saved path through an inner hole to its first opaque disk encounter, and distinguish known darkness from unresolved critical paths."
canonical_url: "https://sandboxphysics.com/stories/accretion-disks/first-opaque-disk-crossing/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/accretion-disks/first-opaque-disk-crossing/"
last_modified: "2026-11-12"
---

L3 · Chapter

# Which disk surface a light path meets first

Sandbox Physics · 2026-11-12

The brightness in one image direction comes from the first emitting, opaque disk surface encountered along that path. A path can pass through the inner hole or cross the disk plane several times. The relevant event is its first actual encounter with gas.

The previous chapter connected emitted radiation to received frequency and intensity. We still need the emitting position. To find it, follow the path back from the observer and let the disk's geometry and opacity select the source.

## Crossing a plane need not mean meeting gas

The thin-disk model places gas in a plane, but only the annulus between its inner and outer edges emits. Inside the inner edge is a hole. Beyond the outer edge, there is no prescribed disk gas.

Think first of looking through a sheet of paper with a hole cut in its middle. Through the hole, your sightline can continue to something behind it. Where it meets paper, you see the paper. A disk calculation needs the same distinction between crossing the plane and encountering opaque material.

Here the disk is optically thick: radiation travelling into it is absorbed over a short path, so the surface blocks direct light from behind it. We use the first surface encountered as the emitting location. [Section 2 of Zimmerman and colleagues' paper](https://arxiv.org/pdf/astro-ph/0408209v1#page=3) states the optically thick premise of the local thermal model. [Li and colleagues' introduction](https://arxiv.org/pdf/astro-ph/0411583v2#page=3) describes tracing paths from the observer toward the disk and calculating the radiation in each image direction.

## A curved path can meet the plane again

Light bends strongly near a black hole. Traced backward, a path can approach the centre and then turn outward. Each time it crosses the disk plane, we must check its radius.

The hole lets a path continue. If the first crossing is inside it and a later crossing reaches an emitting annulus, the later surface supplies the light. Once the path encounters opaque disk material, surfaces farther along it are hidden from direct view.

This is how the far side of a flat disk can appear in unexpected image positions. The disk remains flat; curved paths send light from its different locations into different directions on the observer's sky.

### An actual saved path

Consider the calculated disk with inner radius $9r_g$, where $r_g=GM/c^2$. At a viewing angle of 60 degrees, one saved path first crosses the disk plane at about $6.72r_g$. That crossing lies inside the hole, so the path continues. Its next crossing is at about $10.1r_g$, in the emitting annulus. That is the selected source.

The emitting gas at this crossing has an effective temperature of about 1.47 million kelvin. The frequency ratio to the distant observer is about 0.990. Selection comes first; temperature and motion are then taken from that position, and the radiation is propagated using the previous chapter's relations.

These numbers come from a saved simulated path and its intersection. The accompanying figure uses samples of that same path to show radius along it. It is not a telescope measurement of a gas trajectory. The boundary between hole and emitting region uses the calculation's stated inner radius.

![A saved simulated path crosses the inner hole before meeting the opaque emitting annulus](https://sandboxphysics.com/_astro/first-crossing.en.uFyE95iA.svg)

Figure 2. Path angle orders the backward tracing; the vertical axis gives radius. The curve connects original accepted samples. The first crossing marker evaluates the saved dense polynomial. The view shows radii 3–24 in gravitational units, with the inner-hole portion shaded. Desktop and mobile figures use the same values. [Open the full figure](https://sandboxphysics.com/_astro/first-crossing.en.uFyE95iA.svg)

## A disk encounter comes before the final vacuum fate

A path with no opaque disk encounter has another outcome. Some traced-back paths enter the event horizon. Others pass the black hole and return to the distant region. Against the dark background prescribed here, directions with no disk emission point contribute no disk light.

A vacuum path's eventual capture alone does not make its image direction dark. If it meets the disk first, the image already receives light from that emitting surface. Source selection precedes the path's later vacuum fate.

The size of a dark image region also cannot be read directly as the disk-hole radius. Distance on the image, closest approach of a light path and radius of a disk intersection are different quantities. Bending connects them.

## Why the critical region has a separate marker

Between vacuum escape and capture lies a critical boundary. Paths extremely close to it can spend a long time circling near the black hole. A small initial-direction change can alter the number of turns and later intersections substantially.

A finite calculation must identify which directions are sufficiently determined. Here a very narrow band around the critical boundary remains unresolved, with an upper bound on its possible radiation. The image marks that band separately. The spectrum distinguishes the known contribution from the bounded contribution that may be missing.

Known dark directions and undetermined directions carry different information. The former are a radiation result of the model; the latter need an uncertainty range. A finite image cell can also cover many paths. Its displayed intensity is the area mean over those directions.

## Exercises with explanations

**1.** The example crosses at $6.72r_g$ and $10.1r_g$. Move only the prescribed inner edge from $9r_g$ to $6r_g$. Which crossing now meets the disk first?

The first crossing already lies in the emitting annulus. It supplies light from $6.72r_g$. This compares geometric conditions on the same vacuum path. Temperature and radiation must also be evaluated with the emission model for the chosen inner edge.

**2. A path meets an opaque annulus. Its continued vacuum extension later reaches a hotter annulus. Can their direct intensities be added for that image direction?**

No. The first opaque surface blocks the later direct source. Transparent or partly transparent gas needs a calculation of emission and absorption along the path; adding opaque-surface intensities is not that calculation.

**3. A vacuum path eventually enters the horizon but crosses an emitting disk surface first. Is its direction a case of “no disk light”?**

No. The first disk encounter already selects an emitting location. The later vacuum fate does not erase the direct radiation supplied there.

**4. Transfer problem: a dark image region grows. How would you distinguish a larger inner hole from a region of insufficiently determined paths?**

Inspect the separate unresolved marker, the chosen inner edge, the same sky aperture and the possible-radiation bound. Keeping determined directions separate from the unresolved band makes it possible to distinguish a changed disk hole from numerical uncertainty.

## Sources

1. [Section 2 of Zimmerman and colleagues' paper](https://arxiv.org/pdf/astro-ph/0408209v1#page=3)
2. [Li and colleagues' introduction](https://arxiv.org/pdf/astro-ph/0411583v2#page=3)

## Keep reading

How accretion disks shine · Reading path

### How accretion disks shine

1. [Article **Why gas around a black hole shines**](https://sandboxphysics.com/stories/accretion-disks/why-accretion-disks-shine/)
2. [Article **Where a thin disk heats**](https://sandboxphysics.com/stories/accretion-disks/thin-disk-torque-temperature/)
3. [Article **Adding the spectra of differently heated rings**](https://sandboxphysics.com/stories/accretion-disks/adding-disk-blackbody-spectra/)
4. [Article **How frequency and brightness change on the way out**](https://sandboxphysics.com/stories/accretion-disks/photon-frequency-and-brightness/)
5. [You are here **Which disk surface a light path meets first**](https://sandboxphysics.com/stories/accretion-disks/first-opaque-disk-crossing/)
6. [Article **Recompute a disk image and spectrum from saved intersections**](https://sandboxphysics.com/stories/accretion-disks/recompute-frozen-disk-light/)
7. [Article **How to read a thin-disk spectrum paper**](https://sandboxphysics.com/stories/accretion-disks/reading-thin-disk-spectrum-paper/)

1 more group

### Other articles that use the same experiments

Each experiment from this article that another series also uses, followed by the articles there.

1. [Lab **Black Hole Light Lab**](https://sandboxphysics.com/labs/black-hole-light/)
2. [Article **Why collapse does not need a perfectly spherical star**](https://sandboxphysics.com/stories/astronomy-gravity/penrose-trapped-surfaces/)