Physics

aircraftdetective.utility.physics

_calculate_atmospheric_conditions

_calculate_atmospheric_conditions(altitude)

Computes the air density and temperature as a function of altitute, for altitudes up to 20,000 meters.

All calculations are based on the ISA (International Standard Atmosphere):

Temperature in the Troposphere is calculated according to: $$ T=T_0 - L * h $$ in the lower Stratosphere, it is simply -56.5°C.

where:

Symbol	Dimension	Description	Value
\(T\)	[temperature]	temperature at altitude \(h\)	N/A
\(T_0\)	[temperature]	temperature at sea level	288.15 K
\(L\)	[temperature/length]	temperature lapse rate (Troposphere)	0.0065 K/m
\(h\)	[length]	altitude	N/A

Density is calculated using the exponential approximation: $$ \rho = \rho_0 * e^{-\frac{gh}{RT}} $$ where:

Symbol	Dimension	Description	Value
\(\rho\)	[mass/volume]	air density at altitude \(h\)	N/A
\(\rho_0\)	[mass/volume]	air density at sea level	1.225 kg/m³
\(g\)	[acceleration]	acceleration due to gravity	9.80665 m/s²
\(R\)	[specific gas constant	specific gas constant for air	287 J/(kg*K)
\(T\)	[temperature]	temperature at altitude \(h\)	N/A

Parameters:

Name	Type	Description	Default
altitude	float[distance]	Altitude above sea level	required

Notes

Compare also the function atmos() in openap/extra/aero.py by Junzi Sun. Note that jetfuelburn function has been re-written completely and does not build on the openap code.

References

• Temperature: Eqn.(1.6) in Sadraey (2nd Edition, 2024)
• Density Equations on Wikipedia

Returns:

Type	Description
dict	'density' : ureg.Quantity Air density [kg/m³] 'temperature' : ureg.Quantity Air temperature [°C]

Example

Editor (session: default) Run

import jetfuelburn
from jetfuelburn import ureg
from jetfuelburn.aux.physics import _calculate_atmospheric_conditions
_calculate_atmospheric_conditions(altitude=10000*ureg.m)

Output Clear

Source code in aircraftdetective/utility/physics.py

@ureg.check(
    '[length]' # altitude
)
def _calculate_atmospheric_conditions(altitude: float) -> dict[float, float]:
    r"""
    Computes the air density and temperature as a function of altitute, for altitudes up to 20,000 meters.

    All calculations are based on the ISA (International Standard Atmosphere):

    <img src="https://upload.wikimedia.org/wikipedia/commons/a/a8/International_Standard_Atmosphere.svg" width="250">

    Temperature in the Troposphere is calculated according to:
    $$
        T=T_0 - L * h
    $$
    in the lower Stratosphere, it is simply -56.5°C.

    where:

    | Symbol | Dimension             | Description                         | Value        |
    |--------|-----------------------|-------------------------------------|--------------|
    | $T$    | [temperature]         | temperature at altitude $h$         | N/A          |
    | $T_0$  | [temperature]         | temperature at sea level            | 288.15 K     |
    | $L$    | [temperature/length]  | temperature lapse rate (Troposphere)| 0.0065 K/m   |
    | $h$    | [length]              | altitude                            | N/A          |

    Density is calculated using the _exponential approximation_:
    $$
    \rho = \rho_0 * e^{-\frac{g*h}{R*T}}
    $$
    where:

    | Symbol  | Dimension             | Description                         | Value        |
    |---------|-----------------------|-------------------------------------|--------------|
    | $\rho$  | [mass/volume]         | air density at altitude $h$         | N/A          |
    | $\rho_0$| [mass/volume]         | air density at sea level            | 1.225 kg/m³  |
    | $g$     | [acceleration]        | acceleration due to gravity         | 9.80665 m/s² |
    | $R$     | [specific gas constant| specific gas constant for air       | 287 J/(kg*K) |
    | $T$     | [temperature]         | temperature at altitude $h$         | N/A          |

    Parameters
    ----------
    altitude : float [distance]
        Altitude above sea level

    Notes
    -----
    Compare also the function
    [`atmos()` in `openap/extra/aero.py`](https://github.com/TUDelft-CNS-ATM/openap/blob/39619977962fe6b4a86ab7efbefa70890eecfe36/openap/extra/aero.py#L48C5-L48C10)
    by [Junzi Sun](https://github.com/junzis). Note that `jetfuelburn` function has been re-written completely and does not build on the `openap` code.

    References
    --------
    - Temperature: [Eqn.(1.6) in Sadraey (2nd Edition, 2024)](https://doi.org/10.1201/9781003279068)
    - [Density Equations on Wikipedia](https://en.wikipedia.org/wiki/Barometric_formula#Density_equations)

    Returns
    -------
    dict
        'density' : ureg.Quantity
            Air density [kg/m³]
        'temperature' : ureg.Quantity
            Air temperature [°C]

    Example
    -------
    ```pyodide install='jetfuelburn'
    import jetfuelburn
    from jetfuelburn import ureg
    from jetfuelburn.aux.physics import _calculate_atmospheric_conditions
    _calculate_atmospheric_conditions(altitude=10000*ureg.m)
    ```
    """
    if altitude < 0 * ureg.m:
        raise ValueError("Altitude must not be <0.")
    elif altitude > 20000 * ureg.m:
        raise ValueError("Altitude must not be >20000. We are not considering the stratosphere.")

    temperature_0 = 288.15 * ureg.K # sea-level standard tempreature
    lapse_rate = 0.0065 * (ureg.K/ureg.m) # temperature lapse rate in the troposphere
    temperature_lower_stratosphere = 216.65 * ureg.K # constant temperature in the lower stratosphere
    rho_0 = 1.225 * (ureg.kg/ureg.m ** 3) # sea-level standard atmospheric density
    rho_1 = 0.36391 * (ureg.kg/ureg.m ** 3) # density at 11000 meters
    g = 9.80665 * (ureg.m/ureg.s ** 2) # acceleration due to gravity
    R = 8.3144598 * ((ureg.N * ureg.m)/(ureg.mol*ureg.K)) # univeral gas constant
    M = 0.0289644 * ureg.kg/ureg.mol # molar mass of dry air

    if altitude <= 11000 * ureg.m:
        temperature = temperature_0 - lapse_rate * altitude.to(ureg.m)
        rho = rho_0 * ((temperature_0 - lapse_rate * altitude) / temperature_0) ** (((g * M) / (R * lapse_rate)) - 1)
    else:
        temperature = temperature_lower_stratosphere
        rho = rho_1 * math.exp(-g * M * (altitude - 11000 * ureg.m) / (R * temperature_lower_stratosphere))

    return {
        'density': rho.to(ureg.kg/ureg.m ** 3),
        'temperature': temperature.to(ureg.celsius)
    }