Image 1 Caption: A cartoon of the cosmic distance ladder based on an image in reference FK-587.
The vertical axis is logarithmic distance from the Sun in parsecs: note 1 parsec (pc) = 3.08567758 ... *10**16 m = 206264.806 ... AU = 3.26156377 ... ly ≅ 3.26 ly. The horizontal axis is just to spread out the ranges of applicability of various cosmic distance indicators (distance determination methods) that are used in the cosmic distance ladder.
In this figure/insert, we explicate the cosmic distance ladder.
Features
A cosmic distance indicator is NOT (as usually considered) a direct distance determination method like stellar parallax. It must have a calibration against known calibration standards of known distance (i.e., an empirical calibration) or a theoretical calibration.
Cosmic distance indicators with only theoretical calibration have NOT usually reached sufficient accuracy/precision to compete in accuracy/precision with those relying on empirical calibration. This might change in the future.
Stellar parallax itself is usually NOT considered a cosmic distance indicator since it does NOT rely on calibration. It is a direct distance determination method.
However, stellar parallax is considered to be the first rung of the cosmic distance ladder which is a series of distance determination methods used to reach greater and greater cosmic distances. To construct the cosmic distance ladder each higher rung usually has to calibrated from lower rung. Of course, the first rung (i.e., stellar parallax) does NOT require empirical calibration.
Some rungs can be calibrated using stellar parallax distances.
However, other cosmic distance indicators that reach farther in cosmic distance must be calibrated by others that reach less far.
In fact, the cosmic distance ladder is more like a network than ladder with complicated interrelationships between the rungs.
The vertical axis
is logarithmic
distance from the Sun
in multiples
of the parsec: note
1 parsec (pc) = 3.08567758 ... *10**16 m = 206264.806 ... AU = 3.26156377 ... ly ≅ 3.26 ly.
The horizontal axis
is just to spread out the ranges of applicability of various
cosmic distance indicators
(distance determination methods) that
are used in the
cosmic distance ladder.
As can be seen, the
cosmic distance ladder
is more like a network
than ladder.
Spectroscopic parallax
is explicated in
IAL 19:
Star Basics I: Luminosity Determination and Spectroscopic Parallax.
To explicate, we know to some
accuracy/precision
luminosity L
for some
cosmic distance indicator
(e.g., a main-sequence star
of a given
spectral type
or Type Ia supernovae (SNe Ia))
and then measure
radiant flux (AKA flux) F
for some example of this
cosmic distance indicator
and use the formula
The luminosity L
is the "device" requiring
calibration.
It had to be determined
from representative examples of the
cosmic distance indicator with
known distances r (i.e.,
calibration standards) and
measured fluxes F
via the formula
Of course, there are
uncertainties
in determining
the calibration standard
luminosities L
from the uncertainties
in the determinations of the distances r and
the fluxes F.
Note that for sufficiently remote
astronomical objects,
the astronomical objects
have moved significantly during the light
travel time
for a light signal from them to have
reached us (i.e., the
lookback time)
and that the extinction
is negligible or has been corrected for.
In these cases, the formulae above
do NOT stricly apply and one must do another kind of
calculation
to get a true physical distance.
In the cases when the extinction
is negligible or has been corrected for, the distance r calculated from
r=sqrt[L/(4πF)] is NOT not a true physical distance, but is
referred to as a
luminosity distance.
Luminosity distance
is one of the direct observables that can go into fitting the
free parameters
of the
Λ-CDM model.
For more on luminosity distance,
see file
cosmos_distance_z_10000.html.
Within the
Friedmann equation (FE) models
(which include all conventional
cosmological models),
Hubble's law gives
all cosmological physical distances
(i.e., true physical distances to
astronomical objects
participating in the mean expansion of the universe or, as it is often expressed, participating in the
Hubble flow).
Hubble's law is
Howsoever,
as z
→ 0,
the 1st order
or observational
Hubble's law applies
The
1st order
Hubble's law has the
accuracy as a function of
z
of order the difference between the curve "naive Hubble"
(which is the
1st order
Hubble's law)
curve "LOS comoving"
(which is the
cosmological physical distance)
in the plots
in file
cosmos_distance_z_10000.html.
Circa 2020s, the
standard model of cosmology (SMC, Λ-CDM model)
is the
Λ-CDM model.
The Λ-CDM model may need
revision or replacement sometime soon (maybe within the
2020s), but even so
the cosmological physical distances
it predicts are probably accurate to within ∼ 10%.
The main problem with the
Λ-CDM model
at present is the
Hubble tension.
Direct measurements give H = 74.03(1.42) (km/s)/Mpc and indirect measurements give H = 67.66(42).
The discrepancy is 4.4
standard deviations (i.e., σ's)
which is well beyond statistical probability.
One or other or both of the measurements are in error, but which is the case.
If it is the indirect measurement that is in error,
the Λ-CDM model will have
to be revised or replaced.
Since the discrepancy between the
Hubble constant
values 74.03(1.42) (km/s)/Mpc and 67.66(42) is ∼ 10 %,
the uncertainty
in all distance measurements using the formula
r = (c/H)z = L_H*z is ∼ 10 %.
For more on the
tensions/problems of the
Λ-CDM model,
see file
big_bang_cosmology_limitations.html.
r=sqrt[L/(4πF)]
to determine distance r.
L = (4πr**2)*F .
v = Hr     or, inversely,     r = v/H ,
where r is
cosmological physical distance
and v is
recession velocity.
Alas, neither
cosmological physical distance
nor recession velocity
are direct observables, except
asymptotically
as
cosmological redshift z
→ 0.
zc = Hr     or, inversely,     r = (c/H)z = L_H*z ,
where
vacuum light speed c = 2.99792458*10**8 m/s
(exact by definition) ≅ 3*10**8 m/s = 3*10**5 km/s ≅ 1 ft/ns,
r is
cosmological physical distance,
and
Hubble length L_H = 4.2827 Gpc/h_70 = 13.968 Gly/h_70.
Cepheids,
Hubble constant,
novae,
planetary nebula luminosity function (PNLF),
RR Lyrae variables,
Sigma-D relation
(D_n-σ),
surface brightness fluctuation method (SBF),
Tip of the red-giant branch (TRGB, RGB Tip),
Tully-Fisher relation (TFR),
Type Ia supernovae (SNe Ia),
extragalactic masers.
Cosmic distance ladder videos
(i.e., Cosmic distance ladder
videos):