`Jacchia77.density()` returns value in `kg2 mol / (K m3)`

[Nosudrum]

🐞 Problem

While other atmosphere models COESA62 and COESA76 return density values in kg / m3, Jacchia77 returns a density value in kg2 mol / (K m3).

This can be reproduced with the following code :

from astropy import units as u
from poliastro.earth.atmosphere import coesa62, coesa76, jacchia

# Create atmospheres
atm_coesa62 = coesa62.COESA62()
atm_coesa76 = coesa76.COESA76()
atm_jacchia = jacchia.Jacchia77(1050 * u.K)

# Get densities
alt = 500 * u.km
rho_coesa62 = atm_coesa62.density(alt)
rho_coesa76 = atm_coesa76.density(alt)
rho_jacchia = atm_jacchia.density(alt)
print(f'rho_coesa62 = {rho_coesa62}, rho_coesa76 = {rho_coesa76}, rho_jacchia = {rho_jacchia}')

This generates the following output :

rho_coesa62 = 1.576876104054915e-12 kg / m3, rho_coesa76 = 5.212859332411656e-13 kg / m3, rho_jacchia = 7.346968613514447e-13 kg2 mol / (K m3)

Now, a quick dimensional analysis when reading the documentation of Jacchia77 leads me to think that combining the density output in kg2 mol / (K m3) with jacchia.R in J / (kg mol) and jacchia.k in J / K could lead to something in kg / m3.

However, if that is the expected procedure to get an atmospheric density using Jacchia77 that can then be used in the atmospheric_drag perturbation, then I would suggest to mention it more clearly in the documentation.

If it is not, then I would be interested to know what is and would also suggest updating the Jacchia77 documentation to reflect the fact that its density method does not return rho in usual units, unlike other models.

🖥 Please paste the output of following commands

conda info -a

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conda list

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pip freeze

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jplephem @ file:///home/conda/feedstock_root/build_artifacts/jplephem_1640954253146/work
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[astrojuanlu]

Hi @Nosudrum , thanks for reporting this! There was an attempt to address this #1351 but to be honest I'm slightly confused by this still and we couldn't get the tests to pass. Are you willing to lend us a hand here?

[astrojuanlu]

By the way, the Jacchia77 model is defined here https://ui.adsabs.harvard.edu/abs/1977SAOSR.375.....J/abstract

(PDF link)

[Nosudrum]

Alright, I have identified two problem in the current implementation of the Jacchia77 model in poliastro.earth.atmosphere that are leading to this issue.

Sources

The relevant PDFs are linked here : Jacchia and COESA76.

Density computation and expected units

As you can see, the density computation uses equation (42) of COESA76,

poliastro/src/poliastro/earth/atmosphere/jacchia.py

Lines 146 to 147 in 968ac2c

	# using eqn(42) of COESA76
	rho = P * alt_profile[9] / (R * alt_profile[1])

where alt_profile[9] and alt_profile[1] are respectively the mean molecular weight and temperature returned by altitude_profile.

Equation (42), derived from the ideal gas law and the definition of the molar mass $M$ is :
$$ \rho=\dfrac{P \cdot M}{R\cdot T}$$

The expected units are J/m3 (= Pa) for the pressure $P$, kg/mol for the molar mass $M$, J/(mol.K) for the perfect gas constant $R$ and K for the temperature $T$. This yields a mass density $\rho$ in kg/m3, as expected.

Units in poliastro.earth.atmosphere.jacchia

The units of $P$ and $T$ returned by pressure and altitude_profile are correct.

However, that is not the case for the mean molecular weight WM and $R$ which are returned in kg and J/(mol.kg) respectively.

You can check that yourself with the following code snippet :

from astropy import units as u
from poliastro.earth.atmosphere import jacchia

atm_jacchia = jacchia.Jacchia77(2400 * u.K)
alt = 500 * u.km
R_jacchia = jacchia.R
rho_jacchia = atm_jacchia.density(alt)
P_jacchia = atm_jacchia.pressure(alt)
[z, T, N2, O2, O, Ar, He, H, total_number_density, mean_molecular_weight] = atm_jacchia.altitude_profile(alt)

print(f'P (pressure) = {P_jacchia}')
print(f'WM (mean molecular weight) = {mean_molecular_weight}')
print(f'R (gas constant) = {R_jacchia}')
print(f'T (temperature) = {T}')
print(f'rho (density) = {rho_jacchia}')

which yields

P (pressure) = 1.5321518990161574e-05 J / m3
WM (mean molecular weight) = 18.392271044007167 kg
R (gas constant) = 8314.32 J / (kg mol)
T (temperature) = 2375.6184185930124 K
rho (density) = 1.4267036191401433e-11 kg2 mol / (K m3)

First issue : perfect gas constant R

So there are two issues at hand here, the first with $R$ and the other with $M$.

Let's start with $R$, which is defined here

poliastro/src/poliastro/earth/atmosphere/jacchia.py

Line 10 in 968ac2c

R = 8314.32 * u.J / (u.kg * u.mol)

and only used for the density computation.

My guess is that this line is an interpretation of the definition of $R$ at the bottom of page 8 of the Jacchia PDF. I am not sure if the PDF was wrongly interpreted (the unit of $R$ as described there is not very clear to me either), or if there is a mistake in the PDF. I am enclined to interpret (kg-mol)^-1 as a way to say $R$ might be expressed for a mole or unit mass depending on the situation, but in any case I believe there are units of time missing because kg.m are not the same as J=kg.m2/s2. I am curious about your opinion on this.

Second issue : mean molar weight WM / molar mass M

The unit of the mean molar weight WM returned by altitude_profile is added manually here :

poliastro/src/poliastro/earth/atmosphere/jacchia.py

Line 49 in 968ac2c

WM * (u.kg),

That is clearly a mistake : the sparse air at 500 km is not 18 kg, regardless if it is per molecule or mole. The value is correct though, see the table page 91 of the Jacchia PDF. So there is a problem with the unit.

The mean molecular weight used in the Jacchia PDF is actually unitless (as can be noticed in pages 7 or in the table headers page 91 for example). This is because it is a relative molecular mass (more info here), i.e. the ratio of the mass of a molecule to the unified atomic mass unit. It can basically be treated as equal to the molar mass in g/mol (more info here).

Proposed solutions

For the first issue, related to the perfect gas constant $R$, my proposition would be to simply change the definition of $R$ to fit how it is defined in page 18 of the COESA76 PDF and basically all the other sources available :

R = 8.31432 * u.J / (u.K * u.mol)

For the second issue, the first thing to do is to decide how to handle the mean molar weight properly, i.e. whether to return

• a unitless mean molar weight, which means to remove the * (u.kg) in

poliastro/src/poliastro/earth/atmosphere/jacchia.py

Line 49 in 968ac2c

WM * (u.kg),

• or a molar mass in g/mol or kg/mol with

WM * (u.g / u.mol), # or WM * 1e-3 * (u.kg / u.mol)

The latter is personally my favorite, as the returned unit provides confidence to the user, while a unitless value can lead to misinterpretations. In either cases, it should be properly documented in the API reference to avoid any future confusion.

Additionally, I would suggest adding the following lines (temperature added for code clarity) before computing the density :

T = alt_profile[1]
M = alt_profile[9].to(u.kg / u.mol)

as well as rewriting the equation in a more readable way :

# using eqn(42) of COESA76
 rho = P * M / (R * T)

[astrojuanlu]

Thanks a lot for the super detailed analysis @Nosudrum! 🚀

• I agree with setting R = 8.31432 * u.J / (u.K * u.mol) (which I understand is what Update jacchia.py R gas constant unit  #1351 does)
• I agree with your latter solution for the molecular weigh, i.e. returning a molar mass in u.kg / u.mol (which I understand is what Update jacchia.py R gas constant unit  #1351 is missing) and I definitely agree with your idea to make the equation more readable.

I'll send a quick PR to supersede #1351 and see if it still passes the tests.

[astrojuanlu]

Attempt to fix this issue at #1511