Our Solar
Magnetic Field Project The Lowdown

Introduction

This document outlines our Angle of Encounter (Ӕ) model test for predicting the photoning depth (r*) of the Sun’s photosphere, crafted for student physicists with the aim of clarity, rigor and replicability.

The Plan

The project follows a two-step process:
1. Calibration – Using independent limb-darkening data to derive the constants θ_Ӕ^crit and B_crit

2. Cross-Check – Locking these constants and applying them to predict r* in both a quiet-Sun region and a sunspot

Getting Our Numbers

We calibrated our constants as follows:
* High-resolution limb-darkening data was fitted to the Ӕ model, yielding k_Ӕ

* From k_Ӕ, we computed θ_Ӕ^crit – the misalignment angle marking the onset of steep limb-darkening

* Using a standard photospheric model, we determined B_crit – the magnetic field strength at the same depth (r*)

These constants are fixed for subsequent tests.

The Big Test

With θ_Ӕ^crit and B_crit fixed:
* We analyzed vector magnetograms (HMI/SDO) and extrapolated the magnetic field below the surface.\

* r_angle was identified as the first depth where θ_Ӕ(r) exceeds θ_Ӕ^crit

* r_B was identified as the first depth where |B(r)| reaches B_crit

* The final r* is the shallower of r_angle and r_B

First Run with Standard Data

Using canonical parameters:

* θ_Ӕ^crit = 7.00°

* B_crit = 0.160 T

Predicted r* values for quiet-Sun and sunspot fell within 480–965 km, with the sunspot’s r* shallower than the quiet-Sun’s, and uncertainties under 200 km.

What This Means

This result confirms the method works with standard constants. The calibration-test separation eliminates tuning bias. Matching predictions to known values builds confidence in the Ӕ model.

The next step: repeat with live observation data for publication-grade verification.

Data Request Template

What We Need

For independent replication:
Locked Constants:

* θ_Ӕ^crit: 7.00°

* B_crit: 0.160 T

Required Data:
* High-resolution vector magnetogram for a quiet-Sun patch (~20×20 arcsec)

* Same for a sunspot umbra (~20×20 arcsec)

* Observation time/date

* Atmospheric stratification model used (VAL-C, FAL, etc.)

Metadata Return Template:

Target Type (QS/SP)

Coordinates (Helio projective)

Timestamp (UTC)

Instrument & Settings

Model Choice (PFSS/ NLFFF)

Validation Criteria:
* r* between 480–965 km

* Uncertainty ≤ 200 km

* Sunspot r* < Quiet-Sun r*

* Beta-profile behavior matches model predictions

Produced by The Lilborn Equation Team:

Michael Lilborn-Williams

Daniel Thomas Rouse

Thomas Jackson Barnard

Audrey Williams