Undergrad What is the density of hydrogen atoms in the Universe?

[Buzz Bloom]

TL;DR

I calculated a value of the average density in our universe of H atom equivalents per cm^3 based on values of cosmological variables from various sources: 2.4244x10^18 . A Wikipedia value for such a value in the interstellar media is very very much less: 10^6.
I am hoping a PFs participant can find my error or an explanation.

UNITS

m is meters​

kg is kiliograms​

K is degrees Kelvin​

s is seconds​

J is joules​

u is daltons = 1.66053906660(50)×10−27 kg​

https://en.wikipedia.org/wiki/Dalton_(unit)​

1 pc = 3.085678 x 10¹⁶ m​

CONSTANTS

M_H = mass of hydrogen atom = 1.007825 u​

= 1.673532784796145 ×10⁻²⁷ kg​

https://en.wikipedia.org/wiki/Hydrogen_atom​

G = gravitational constant = 6.6743 ×10⁻¹¹ m³⋅/kg⋅s²​

VARIABLES

H₀ = 70 (km/s)/Mpc​

https://en.wikipedia.org/wiki/Hubble's_law#Measured_values_of_the_Hubble_constant​

(This reference has an array of values for H₀, each with an error range. The value 70 was calculated as a weighted average using the inverse of the square of the error ranges as multipliers.)​

1/H₀ = 4.408 x 10¹⁷ s​

​

DENSITY CALCULATION

Ω_m = 0.315 (total mass ratio to critical mass))​

https://ui.adsabs.harvard.edu/abs/2020A&A...641A...6P/abstract​

Ω_b = 0.15 x 0.315 = 0.04725 (baryon mass ratio)​

(I apologize for losing the URL from which I found 0.15)​

ρ_c = critical density = 3H²/8πG​

https://en.wikipedia.org/wiki/Friedmann_equations#Density_parameter​

ρ_b = Ω_b x ρ_c​

= (3/8pi) H₀²/G = 4.0573 x 10^-9 kg/m³​

N_H = current density of H atom equivalents =ρ_b / M_H​

= 2.4244 x 10¹⁸ per m³​

​

DENSITY FROM WIKIPEDIA

matter is primarily in molecular form, and reaches number densities of 10⁶ molecules per cm³​

https://en.wikipedia.org/wiki/Interstellar_medium​

​

 

[mathman]

You have per $m^3$. Wiki has per $cm^3$. factor is $10^6$.

 

[PeterDonis]

matter is primarily in molecular form

You're calculating the average density of hydrogen atoms over the entire universe, which includes the volume occupied by things like stars, planets, white dwarfs, and neutron stars, which can have densities many, many orders of magnitude larger than average, and also includes volumes like the "voids" between galactic superclusters, which can have densities orders of magnitude smaller than average.

You are then comparing this to an estimate of the density of the interstellar medium. You should not expect these two numbers to be the same (even after correcting for the units as @haruspex says).

 

[Buzz Bloom]

Hi @mathman and @PeterDonis:

I much appreciate your helpful posts. My faulty intuition, even with mathman's correction, was that the variability of average density is much less than it apparently is.

Regards,
Buzz