Talk:Observable universe/workpage

Mass of Ordinary Matter

The mass of the universe is often quoted as 10⁵⁰ tons or 10⁵³ kg. In this context, mass refers to ordinary matter which includes Interstellar Medium (ISM) and Intergalactic Medium (IGM); however, it excludes dark matter and dark energy. Three calculations substantiate this quoted value for the mass of ordinary matter in the universe: Estimating based on critical density; Extrapolating from number of stars; and, Estimating based on steady-state. The calculations obviously assume a finite universe.

Estimating based on critical density

Critical Density is the energy density where the expansion of the universe is poised between continued expansion and collapse. Observations of the cosmic microwave background from the Wilkinson Microwave Anisotropy Probe suggest that the spatial curvature of the universe is very close to zero, which in current cosmological models implies that the value of the density parameter must be very close to a certain critical density value. At this condition, the calculation for $\rho _{c}$ critical density, is):

$\rho _{c}={\frac {3H_{0}^{2}}{8\pi G}}$

where G is the gravitational constant. From The European Space Agency’s Planck Telescope results: $H_{0}$, is 67.15 kilometers per second per mega parsec. This gives a critical density of 0.85 x 10^-26 km/m³ (commonly quoted as about 5 hydrogen atoms/m³). This density includes four significant types of energy/mass: ordinary matter (4.8%), neutrinos (0.1%), cold dark matter (26.8%), and dark energy (68.3%). Note that although neutrinos are defined as particles like electrons, they are listed separately because they are difficult to detect and so different from ordinary matter. Thus, the density of ordinary matter is 4.8% times the total critical density calculated or 4.08 x 10^-28 km/m³. To convert this density to mass we must multiply by volume, a value based on the radius of the "observable universe". Since the universe has been expanding for 13.7 billion years, the comoving distance (radius) is now about 46.6 billion light years. Thus, volume (4/3 π r³) equals 3.58 x 10⁸⁰ m³ and mass of ordinary matter equals density (4.08 x ^-28 km/m³) times volume (3.58 x 10⁸⁰ m³) or 1.46 x 10⁵³ kg.

Extrapolating from number of stars

Obviously there is no way to know exactly the number of stars, but from current literature, the range of 10²² to 10²⁴ is normally quoted. One way to substantiate this range is to estimate the number of galaxies and multiply by the number of stars in an average galaxy. The 2004 Hubble Ultra Deep Field image contains an estimated 10,000 galaxies. The patch of sky in this area, is 3.4 arc minutes on each side. For a relative comparison, it would require over 50 of these images to cover the full moon. If this area is typical for the entire sky, there are over 100 billion galaxies in the universe. More recently, in 2012, Hubble scientists produced the “eXtreme Deep Field” image which showed slightly more galaxies for a comparable area. However, in order to compute the number of stars based on these images, we would need additional assumptions: the percent of both large and dwarf galaxies; and, their average number of stars. Thus, a reasonable option is to assume 100 billion average galaxies and 100 billion stars per average galaxy. This results in 10 ²² stars. Next, we need average star mass which can be calculated from the distribution of stars in the Milky Way. Within the Milky Way, if a large number of stars are counted by spectral class, 73% are class M stars which contain only 30% of the Sun’s mass. Considering mass and number of stars in each spectral class, the average star is 51.5% of the Sun’s mass. The Sun’s mass is 2 x 10 ³⁰ kg. so a reasonable number for the mass of an average star in the universe is 10³⁰ kg. Thus, the mass of all stars equals the number of stars (10²²) times an average mass of star (10³⁰ kg) or 10⁵² kg . The next calculation adjusts for Interstellar Medium (ISM) and Intergalactic Medium (IGM). ISM is material between stars: gas (mostly hydrogen) and dust. IGM is material between galaxies, mostly hydrogen. Ordinary matter (protons, neutrons and electrons) exists in ISM and IGM as well as in stars. In the reference, "The Cosmic Energy Inventory“, the percentage of each part is defined: stars - 5.9%, Interstellar Medium (ISM) - 1.7%, and Intergalactic Medium (IGM) - 92.4%. Thus, to extrapolate the mass of the universe from the star mass, divide the 10⁵⁵ kg mass calculated for stars by 5.9%. The result is 1.7 x 10⁵³ kg for all the ordinary matter.

Estimating based on steady-state universe

Sir Fred Hoyle calculated the mass of an observable steady-state universe using the formula:

${\frac {4}{3}}\cdot \pi \cdot \rho \cdot \left({\frac {c}{H}}\right)^{3}$

which can also be stated as

${\frac {c^{3}}{2GH}}\$

Here H = Hubble constant, ρ = Hoyle's value for the density, G = gravitational constant and c = speed of light. This calculation yields approximately 0.9 × 10⁵³ kg; however, this represents all energy/matter and is based on the Hubble volume, which is the volume of a sphere with radius equal to the Hubble length or about 13.7 billion light years. Thus, to compare with the critical density calculation , the larger volume based on the 46.6 billion light years radius must be used and the critical density for all energy/matter must be multiplied by volume. The comoving distance radius gives a volume about 39 times greater (46.7 cubed divided by 13.7 cubed). So multiplying the calculated mass, 0.9 x 10⁵³ kg, times 39 is the value to compare . The Hoyle result is thus 3.5 x 10⁵⁴ kg for all energy/mass. As noted above in the "Estimating based on critical density", ordinary matter is 4.8% of all energy/matter. If the Hoyle result is multiplied by this percent, the result for ordinary matter is 1.68 x 10⁵³ kg. In summary, the three independent calculations produced reasonably close results :1.46 x 10⁵³ kg, 1.7 x 10⁵³ kg , and 1.68 x 10⁵³ kg. The average is 1.6 x 10⁵³ kg. The key assumptions using the Extrapolation from Star Mass method were number of stars (10²²) and percent of ordinary matter in stars (5.9%). The key assumptions using Critical Density were radius of universe (46.6 billion light years) and percent of ordinary matter in all matter (4.8 %). Both Critical Density and the Hoyle Steady-state equation also used the Hubble constant (67.15 km/sec/Mpc).

Matter content - Number of Atoms

Assuming the mass of ordinary matter is about 1.6 x 10⁵³ kg (reference previous section) and assuming all atoms are hydrogen atoms (which in reality make up about 74% of all atoms in our galaxy by mass, see Abundance of the chemical elements), calculating the total number of atoms in the universe is straight forward. Divide the mass of ordinary matter by the mass of a hydrogen atom (1.6 x 10⁵³ kg divided by 1.67 x 10^-27 kg). The result is approximately 10⁸⁰ hydrogen atoms. Jim Johnson 22:29, 26 July 2013 (UTC)