AIRMASS

This function computes atmospheric air mass, which is the integrated mass of air per unit surface area along a line of sight through the atmosphere. It is commonly used in atmospheric science, optics, and solar engineering to quantify how much atmosphere a path traverses at a given viewing angle.

The model integrates density with geometric and refractive corrections as a function of elevation:

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

The density profile is supplied as tabulated height-density points and linearly interpolated between points. The angle is measured above the horizon, where 90^\circ corresponds to a vertical path with minimum air mass.

Excel Usage

=AIRMASS(density_profile, angle, h_max, r_planet, ri)

density_profile (list[list], required): 2D array with elevation (m) in the first column and density (kg/m³) in the second column
angle (float, required): Angle above the horizon in degrees (0-90)
h_max (float, optional, default: 86400): Maximum height to integrate to (m)
r_planet (float, optional, default: 6371229): Radius of the planet (m)
ri (float, optional, default: 1.000276): Refractive index of the atmosphere

Returns (float): Mass of air per square meter (float), or error message string.

Example 1: Zenith air mass (straight up)

Inputs:

density_profile		angle
0	1.225	90
1000	1.112

Excel formula:

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

Expected output:

96133.3

Example 2: Air mass at 45 degree angle

Inputs:

density_profile		angle	h_max	r_planet
0	1.225	45	86400	6371229
1000	1.112

Excel formula:

=AIRMASS({0,1.225;1000,1.112}, 45, 86400, 6371229)

Expected output:

135059

Example 3: All arguments specified explicitly

Inputs:

density_profile		angle	h_max	r_planet	ri
0	1.225	90	86400	6371229	1.000276
1000	1.112

Excel formula:

=AIRMASS({0,1.225;1000,1.112}, 90, 86400, 6371229, 1.000276)

Expected output:

96133.3

Example 4: Custom planet radius

Inputs:

density_profile		angle	h_max	r_planet	ri
0	1.225	90	86400	7000000	1.000276
1000	1.112

Excel formula:

=AIRMASS({0,1.225;1000,1.112}, 90, 86400, 7000000, 1.000276)

Expected output:

96133.3

Python Code

Show Code

from fluids.atmosphere import airmass as fluids_airmass

def airmass(density_profile, angle, h_max=86400, r_planet=6371229, ri=1.000276):
    """
    Calculate the mass of air per square meter in the atmosphere along a given angle using a density profile.

    See: https://fluids.readthedocs.io/fluids.atmosphere.html

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

    Args:
        density_profile (list[list]): 2D array with elevation (m) in the first column and density (kg/m³) in the second column
        angle (float): Angle above the horizon in degrees (0-90)
        h_max (float, optional): Maximum height to integrate to (m) Default is 86400.
        r_planet (float, optional): Radius of the planet (m) Default is 6371229.
        ri (float, optional): Refractive index of the atmosphere Default is 1.000276.

    Returns:
        float: Mass of air per square meter (float), or error message string.
    """
    # Helper to normalize 2D list input
    def to2d(x):
        return [[x]] if not isinstance(x, list) else x

    try:
      # Normalize density_profile input
      density_profile = to2d(density_profile)

      # Validate density_profile
      if not isinstance(density_profile, list) or len(density_profile) < 2:
        return "Error: density_profile must be a 2D list with at least two rows."
      if not all(isinstance(row, list) and len(row) >= 2 for row in density_profile):
        return "Error: density_profile must contain rows with at least two columns."
      try:
        heights = [float(row[0]) for row in density_profile]
        densities = [float(row[1]) for row in density_profile]
      except Exception:
        return "Error: density_profile must contain numeric values."
      if any(h2 <= h1 for h1, h2 in zip(heights, heights[1:])):
        return "Error: heights in density_profile must be strictly increasing."

      try:
        gamma = float(angle)
        h_max_val = float(h_max)
        r_planet_val = float(r_planet)
        ri_val = float(ri)
      except Exception:
        return "Error: angle, h_max, r_planet, and ri must be numbers."

      if not (0 <= gamma <= 90):
        return "Error: angle must be between 0 and 90 degrees."
      if h_max_val <= 0:
        return "Error: h_max must be positive."
      if r_planet_val <= 0:
        return "Error: r_planet must be positive."
      if ri_val <= 0:
        return "Error: ri must be positive."

      # 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]

      result = fluids_airmass(density_func, gamma, h_max_val, r_planet_val, ri_val)
      return result
    except Exception as e:
      return f"Error: {str(e)}"

Online Calculator

density_profile *

2D array with elevation (m) in the first column and density (kg/m³) in the second column

angle *

Angle above the horizon in degrees (0-90)

h_max

Maximum height to integrate to (m)

r_planet

Radius of the planet (m)

Refractive index of the atmosphere