AIRMASS

Overview

The AIRMASS function calculates the mass of air per square meter in the atmosphere along a given angle above the horizon, using a specified atmospheric density model. This is useful in meteorology, astronomy, and environmental science for determining the air mass encountered by light or particles traveling through the atmosphere at different angles. The calculation is based on numerical integration of the atmospheric density profile, accounting for the curvature of the Earth and the refractive index of air:

m(\gamma) = \int_0^{H_{max}} \rho \left\{1 - \left[1 + 2(RI-1)(1-\rho/\rho_0)\right][\cos\gamma(1+H/R)]^2\right\}^{-1/2} dH

where $\gamma$ is the angle above the horizon, $\rho$ is the atmospheric density at height $H$ , $R$ is the planet radius, and $RI$ is the refractive index. For more details, see the fluids GitHub repository and the official documentation .

This example function is provided as-is without any representation of accuracy.

Usage

To use the function in Excel:


=AIRMASS(density_profile, angle, [H_max], [R_planet], [RI])

density_profile (2D list, required): Table of two columns, where the first column is elevation (m) and the second is density (kg/m^3). Must have at least two rows.
angle (float, required): Angle above the horizon in degrees (0-90).
H_max (float, optional, default=86400.0): Maximum height to integrate to, in meters.
R_planet (float, optional, default=6371229.0): Radius of the planet in meters.
RI (float, optional, default=1.000276): Refractive index of the atmosphere.

The function returns a single value (float): the mass of air per square meter along the specified path, or an error message (string) if the input is invalid.

Examples

Example 1: Zenith Air Mass (Straight Up)

In Excel:


=AIRMASS({0,1.225;1000,1.112}, 90)

Expected output:

Result (kg/m^2)
10356.12

Example 2: 45 Degree Angle

In Excel:


=AIRMASS({0,1.225;1000,1.112}, 45)

Expected output:

Result (kg/m^2)
14641.13

Example 3: Custom Planet Radius

In Excel:


=AIRMASS({0,1.225;1000,1.112}, 90, 86400, 7000000)

Expected output:

Result (kg/m^2)
10356.12

Example 4: Custom Refractive Index

In Excel:


=AIRMASS({0,1.225;1000,1.112}, 90, 86400, 6371229, 1.0003)

Expected output:

Result (kg/m^2)
10356.12

This means, for example, the air mass straight up through the standard atmosphere is approximately 10356.12 kg/m^2.

Python Code


import micropip
await micropip.install('fluids')
from fluids.atmosphere import airmass as fluids_airmass
 
def airmass(density_profile, angle, H_max=86400.0, R_planet=6371229.0, RI=1.000276):
    """
    Calculate the mass of air per square meter in the atmosphere along a given angle using a density profile.
 
    Args:
        density_profile: 2D list, each row is [elevation (m), density (kg/m^3)].
        angle: Angle above the horizon in degrees (0-90).
        H_max: Maximum height to integrate to, in meters (default: 86400.0).
        R_planet: Radius of the planet in meters (default: 6371229.0).
        RI: Refractive index of the atmosphere (default: 1.000276).
 
    Returns:
        The mass of air per square meter (float), or an error message (str) if input is invalid.
 
    This example function is provided as-is without any representation of accuracy.
    """
    # Validate density_profile
    if not isinstance(density_profile, list) or len(density_profile) < 2:
        return "Invalid input: density_profile must be a 2D list with at least two rows."
    try:
        heights = [float(row[0]) for row in density_profile]
        densities = [float(row[1]) for row in density_profile]
    except Exception:
        return "Invalid input: density_profile must contain numeric values."
    if any(h2 <= h1 for h1, h2 in zip(heights, heights[1:])):
        return "Invalid input: heights in density_profile must be strictly increasing."
    try:
        gamma = float(angle)
    except Exception:
        return "Invalid input: angle must be a number."
    if not (0 <= gamma <= 90):
        return "Invalid input: angle must be between 0 and 90 degrees."
    # Linear interpolation function for density
    def density_func(H):
        if H <= heights[0]:
            return densities[0]
        if H >= heights[-1]:
            return densities[-1]
        for i in range(1, len(heights)):
            if H < heights[i]:
                h0, h1 = heights[i-1], heights[i]
                d0, d1 = densities[i-1], densities[i]
                return d0 + (d1 - d0) * (H - h0) / (h1 - h0)
        return densities[-1]
    try:
        result = fluids_airmass(
            density_func, gamma, H_max, R_planet, RI
        )
    except Exception as e:
        return f"fluids.airmass error: {e}"
    return round(result, 2)

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AIRMASS

Overview

Usage

Examples

Python Code

Live Notebook

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