Model Validation & Photometric Formulas | Visual Magnitude Calculator

Photometric Reference

1. Stellar Visual Magnitude Baseline

Visual magnitude ($m_v$) is a logarithmic scale representing apparent brightness as viewed from Earth. The baseline reference is calibrated using standard solar flux and bright stellar benchmarks such as the Orion constellation.

Orion Constellation Visual Magnitude Map

Figure 1: Visual magnitude calibration scale and Orion constellation reference stars (Tap to enlarge).

The standard visual magnitude formula relating received irradiance $E$ to a standard reference $E_0$ is expressed as:

$$m_v = -2.5 \log_{10}\left(\frac{E}{E_0}\right)$$

Target Physics

2. Target Photometry & Lambertian Sphere Model

Calculating the visual magnitude of an artificial satellite requires accounting for range, cross-sectional area, material albedo, and sun-target-observer geometry.

Figure 2: The four core parameters determining satellite visual magnitude (Tap to enlarge).

Lambertian Sphere Illumination Model

Assuming a diffuse spherical satellite (Lambertian surface) of radius $r$, cross-sectional area $A = \pi r^2$, surface reflectivity $\rho$, at range $R$, and phase angle $\phi$ (angle between Sun-Satellite and Satellite-Observer vectors):

$$F(\phi) = \frac{2}{3\pi} \left[ \sin\phi + (\pi - \phi)\cos\phi \right]$$

The calculated solar irradiance reflected toward the observer $E_{\text{sat}}$ is given by:

$$E_{\text{sat}} = E_{\text{sun}} \cdot \rho \cdot \frac{A}{R^2} \cdot F(\phi)$$

Where $E_{\text{sun}}$ is the exo-atmospheric solar irradiance constant in the visible waveband.

Optical Payload Modeling

3. Sensor Payload & Photon Capture Model

Once the target irradiance at the aperture is known, the optical sensor model estimates total detected photons and system angular resolution.

Figure 3: Key optical sensor parameters affecting image resolution and photon capture (Tap to enlarge).

Total Photon Capture Calculation

The total collected photons $N_{\text{photon}}$ received during integration time $t_{\text{int}}$ across aperture diameter $D$, optical throughput $\tau$, and detector quantum efficiency $QE$ is calculated as:

$$N_{\text{photon}} = \left( \frac{E_{\text{sat}}}{h \nu} \right) \cdot \left( \pi \frac{D^2}{4} \right) \cdot t_{\text{int}} \cdot \tau \cdot QE$$

Where $h$ is Planck's constant and $\nu$ is the mean central optical frequency.

Angular Resolution & Spatial Metrics

Diffraction Limit (Rayleigh Criterion): $\theta = 1.22 \frac{\lambda}{D}$ (radians)
Detector Instantaneous Field of View (IFOV): $\text{IFOV} = \frac{p}{f}$ (radians), where $p$ is detector pitch/size and $f$ is focal length.

App Implementation

4. Wavelength Adjustments & Results Export

Figure 4: Customizing spectral bands, unit selections, and analysis reporting (Tap to enlarge).

The Visual Magnitude Calculator allows real-time adjustments across custom spectral passbands (e.g., Visible, Near-IR), unit toggles, and snapshot logging to support field reporting and space domain awareness research.