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Rainbow formation: from Descartes to Venus

This paper analyzes the geometric optics of rainbow formation based on Descartes' and Newton's theories to calculate the angular dimensions of primary and secondary rainbows on Venus, demonstrating that the concentration of aqueous sulfuric acid droplets most significantly affects the size of Alexander's dark band.

Original authors: Andrey Zaikin

Published 2026-03-18

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Andrey Zaikin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Rainbow Recipe: From Earth's Rain to Venus's Acid Clouds

Imagine a rainbow not just as a pretty picture in the sky, but as a giant, natural prism machine. This paper by A.D. Zaikin is like a detective story where the author takes the classic recipe for making a rainbow on Earth and asks: "What happens if we swap the ingredients?"

Here is the breakdown of the science, translated into everyday language.

1. The Classic Earth Recipe: The Water Droplet

On Earth, rainbows happen because sunlight hits a spherical water droplet, bounces around inside, and shoots back out toward your eye.

The "Bounce" (Reflection): Think of the droplet as a tiny, transparent billiard ball. When a beam of light hits it, it doesn't just pass through; it gets trapped, bounces off the back wall, and zooms out the front.
The "Sweet Spot" (The Rainbow): Not every light ray comes out at the same angle. Most rays scatter in random directions, but there is a specific "sweet spot" where the light rays bunch up tightly, like a crowd of people squeezing through a narrow door. This bunching creates the bright, colorful arc we see.
The Double Bounce (Secondary Rainbow): Sometimes, the light bounces twice inside the drop before escaping. This creates a second, fainter rainbow outside the first one.
The Dark Gap (Alexander's Band): Between the primary and secondary rainbows, there is a dark strip of sky. Why? Because no light rays are bouncing out at those specific angles. It's like a "no-fly zone" for light.

The Newton Twist:
Isaac Newton discovered that white light is actually a mixture of colors (a rainbow in a bottle). Because different colors bend (refract) at slightly different angles, the "sweet spot" for red light is slightly different from the "sweet spot" for violet light. This spreads the light out into the spectrum we love.

2. The Venus Experiment: Swapping Water for Acid

Now, let's travel to Venus. The clouds there aren't made of water; they are made of sulfuric acid.

The author asks: If we replace the water droplets with sulfuric acid droplets, how does the rainbow change?

To answer this, he uses Geometric Optics (basically, drawing lines to see where light goes) and changes the "refractive index."

The Refractive Index: Think of this as the "stickiness" of the material to light. Water has a certain stickiness. Sulfuric acid is "stickier" (it bends light more).
The Variable: The clouds on Venus aren't all the same. Some droplets are mostly water with a little acid; others are almost pure acid. The author calculated what happens as the "acid concentration" goes from 0% (pure water) to 95% (super concentrated acid).

3. The Big Discovery: The "Gap" Explodes

Here is the most surprising part of the paper.

When you increase the concentration of sulfuric acid in the droplets:

The Primary Rainbow (Inner Arc): It actually gets smaller. It shrinks closer to the center.
The Secondary Rainbow (Outer Arc): It gets bigger. It expands outward.
The Dark Gap (Alexander's Band): This is the big winner. As the acid gets stronger, the gap between the two rainbows explodes in size.

The Analogy:
Imagine the two rainbows are two rubber bands stretched around a ball.

On Earth (water), the bands are close together, with a small gap of about 7 degrees between them.
On Venus (high acid), the inner band shrinks and the outer band expands. The gap between them stretches out to over 45 degrees.

It's like stretching a rubber band until the space in the middle is huge.

4. Why Does This Matter? (The Detective Work)

Why do we care about the size of the dark gap?

Because the gap is a ruler.

If a spacecraft (like the VenusExpress mentioned in the paper) flies over Venus and sees a rainbow, the scientists can measure the size of the dark gap between the arcs.

Small gap? The clouds are mostly water.
Huge gap? The clouds are very concentrated sulfuric acid.

This allows scientists to "taste" the clouds without ever landing a probe in them. They can diagnose the chemical composition of the Venusian atmosphere just by looking at the size of the rainbow's dark belly.

Summary

This paper takes the classic physics of rainbows (which we learned in school) and applies it to an alien world. It shows that while the mechanism of the rainbow is the same everywhere (light bouncing in drops), the size of the dark gap is a super-sensitive indicator of what the drops are made of.

On Venus, a rainbow isn't just a pretty sight; it's a chemical signpost telling us exactly how much acid is in the sky.

1. Problem Statement

The paper addresses the optical mechanics of rainbow formation, extending classical terrestrial theories (Descartes and Newton) to extraterrestrial environments, specifically the atmosphere of Venus. While the formation of rainbows on Earth (via water droplets) is well-understood through geometric optics and wave theory, the specific parameters of rainbows formed by sulfuric acid droplets in Venus's clouds remain less explored in terms of quantitative angular characteristics. The primary objective is to determine how the concentration of sulfuric acid in Venusian cloud droplets affects the angular dimensions of primary and secondary rainbows and the intervening "Alexander's dark band."

2. Methodology

The study employs geometric optics to model light scattering, a method chosen because it accurately describes the angular positions of rainbow extrema (intensity peaks) without the computational complexity of full electromagnetic wave scattering (Mie theory).

Theoretical Framework:
- Single Reflection (Primary Rainbow): The path of a light ray entering a spherical droplet, refracting, reflecting once internally, and refracting out is modeled. The angle of deviation ( $\gamma$ ) is derived as a function of the angle of incidence ( $\alpha$ ) and the refractive index ( $n$ ). The maximum deviation angle is found by setting the derivative $d\gamma/d\alpha = 0$ .
- Double Reflection (Secondary Rainbow): A similar derivation is performed for rays undergoing two internal reflections.
- Dispersion: The study incorporates the dependence of the refractive index on wavelength (dispersion) using the Cauchy equation to explain the separation of colors (violet vs. red).
Application to Venus:
- The model assumes Venusian clouds consist of spherical droplets of aqueous sulfuric acid.
- Refractive index data for sulfuric acid solutions at various mass concentrations (0% to 95.6%) and wavelengths (360 nm to 714 nm) are sourced from experimental literature (Palmer & Williams).
- Polynomial approximations (4th order) are used to relate the refractive index to the mass fraction of sulfuric acid ( $\omega$ ).
Calculations:
- The authors calculate the angles of maximum intensity ( $\gamma_{01}$ for primary, $\gamma_{02}$ for secondary) for red and violet light across a range of acid concentrations.
- The angular width of Alexander's dark band (the region between the primary and secondary rainbows where no light is scattered) is calculated as $\Delta = \gamma_{02} - \gamma_{01}$ .

3. Key Contributions

Extension of Classical Theory: The paper successfully adapts the Descartes-Newton geometric optics framework to non-aqueous, high-refractive-index media (sulfuric acid).
Quantitative Modeling of Venusian Rainbows: It provides the first detailed calculation of how varying sulfuric acid concentrations alter the specific angular geometry of rainbows on Venus.
Identification of a Diagnostic Tool: The study identifies the angular size of Alexander's dark band as the most sensitive parameter to changes in cloud composition, offering a potential remote sensing method.

4. Key Results

The calculations reveal significant deviations from Earth-based rainbow parameters as sulfuric acid concentration increases:

Primary Rainbow ( $\gamma_{01}$ ): As the mass fraction of sulfuric acid increases from 0% (water) to 90%, the angular radius of the primary rainbow for red light decreases significantly (from ~42°36' to ~29°13').
Secondary Rainbow ( $\gamma_{02}$ ): Conversely, the angular radius of the secondary rainbow increases with concentration (from ~49°57' to ~74°35').
Alexander's Dark Band: The most dramatic effect is observed in the dark band separating the two rainbows. Its angular width expands by more than six times, growing from approximately 7°20' (for water) to 45°23' (for high-concentration acid solutions).
Color Dispersion: The separation between red and violet rays remains present but shifts in absolute angular position based on the refractive index of the specific acid concentration.
Polynomial Models: The authors provide specific 4th-order polynomial coefficients to approximate the refractive index and the Alexander's band width as functions of acid mass fraction, enabling rapid estimation for remote sensing applications.

5. Significance

Remote Sensing of Venus: The findings suggest that by observing the angular size of the dark band between rainbow arcs in Venusian clouds, scientists can remotely diagnose the concentration of sulfuric acid in the cloud droplets without direct sampling.
Planetary Atmosphere Characterization: This work contributes to the broader understanding of optical phenomena in extraterrestrial atmospheres, distinguishing between phenomena caused by refraction (rainbows) and diffraction (glories), the latter of which was recently observed on Venus.
Educational and Theoretical Value: The paper reinforces the utility of geometric optics in analyzing complex atmospheric scattering problems and bridges historical optical theories (Descartes/Newton) with modern planetary science.

In conclusion, Zaikin demonstrates that the "rainbow" on Venus would look fundamentally different from Earth's, characterized by a much wider dark gap between the primary and secondary arcs, a feature directly correlated to the chemical composition of the clouds.