- 4006 - GALACTIC DISTANCES - how do we measure them? Measuring distances in astronomy is a very difficult process. Many of the prior reviews have covered specific methods in some detail. In this review I have put several methods all together to step by step reach galaxies at the edge of the Universe.
-------------- 4006 - GALACTIC DISTANCES - how do we measure them?
- I have used half a dozen of the over two
dozen distance measuring methods used by astronomers. We start with the simplest. Plain trigonometry.
-
- OUT TO 200 LIGHTYEARS: We start by measuring the angle to a nearby
star in relation to its more distant background stars. Then six months later when the Earth is on
the other side of the Sun we measure the angle again. The angle will change.
-
- The nearby star will have shifted relative
to its background stars due to parallax. The same occurs when you extend your
hand and view you thumb with one eye then the other and the background scenery
shifts due to parallax.
-
- The sine of the parallax angle = the
opposite side / hypotenuse. The opposite
side of the right triangle is the distance between the Sun and the Earth and
the hypotenuse is the distance to the star.
-
- PARALLAX EXAMPLE : The closest star is Proxima Centauri. The six month parallax angle was measured to
be 0.76 arc seconds. There are 3600
arc-seconds in 1 degree. 0.76
arc-seconds = 2.11*10^-4 degrees. The
sine of the parallax angle is 3.685*10^-6
The distance from the Sun to the Earth is 1.5*10^11 meters.
-
- Distance to Proxima Centauri =
(1.5*10^11) / (3.685*10^-6)
= 4.1*10^16 meters
-
--------------- 1 lightyear = 9.46*10^15 meters
-
------------------ Distance to Proxima Centauri = 4.3
lightyears, our closest star.
-
- Next we measure the brightness of the star in
watts / meter^2 (F) where we know the distance (D). Now we can calculate the intrinsic luminosity
(L) of the star.
-
--------------------- L
= 4*pi*D^2 * F
-
--------------------- where:
4*pi*D^2 is the surface area of a sphere of radius D
-
-------------------- where:
“F” is the Flux, or apparent luminosity, or brightness of the star that
we measured.
-
------------------- where:
“L” is intrinsic luminosity of the star.
-
----------------- LUMINOSITY EXAMPLE: L
= 4*pi*D^2 * F
-
------------------ where:
F is the Flux we measured to be 1*10^-12 watts/m^2
-
------------------ where L = 3.8*10^26 watts, the same
luminosity as our Sun.
-
----------------- therefore: D
= 5.5*10^18 meters, or 581
lightyears
-
- OUT TO 100,000 LIGHTYEARS: Next we find a star in a more distant galaxy
that is similar, and we can assume that it has the same intrinsic luminosity. We measure its Brightness, F and calculate
the distance D using the same formula:
-
---------------------- D^2 =
L / 4*pi*F
-
- Next, rather than use a single star that we
think is similar we measure the brightness of dozens of stars and plot them
versus their temperature, or color. This
plot is called the H-R diagram, the
Hertzsprung - Russell diagram.
-
- Luminosity versus temperature plots the
stars of various masses that are on the Main-Sequence which is a straight line
on this H-R graph. Luminosities range
from 10^-3 to 10^5 Solar Luminosities.
Temperatures range from 30,000 Kelvin to 3,000 Kelvin, corresponding to
colors from Blue to Red.
-
- The size of the stars range from 0.1 to 10 Solar Radius. We do this straight line plot for stars in a
galaxy where we know the distance (Hyades Cluster). Next, we create a similar
plot for stars in a galaxy where we want to find the distance (Pleiades
Cluster).
-
- Wien’s law is used to measure temperature
by measuring the wavelength of maximum brightness. Temperature = 00029 m*K / wavelength
max. Now we have two straight line
Main-Sequence plots that shift in dimness for the more distant galaxy. The distance squared is equal to the shift in
brightness.
-
- STAR CLUSTER EXAMPLE: The Hyades Cluster
of stars can be measured using the parallax method to have a distance of 151
lightyears. We would like to calculate
the distance to the Pleiades Cluster of Stars know as the Seven Sisters. We measure
the brightness and temperature of 50 stars in the Hyades Cluster and plot a
Main-Sequence line.
-
- The relative brightness ranges from 0.1 to
50. The temperature ranges from 4000 to
8000 Kelvin. Next, we do the same thing
for 50 stars in the Pleiades Cluster.
The Main-Sequence straight line is shifted to 7.5 times dimmer than the
straight line for the Hyades Cluster.
That means that it is the square root of 7.5 further distant away. 2.75 times 151 lightyears = 415 lightyears
away for the Pleiades Cluster.
-
- OUT TO 100,000,000 LIGHTYEARS: Among the stars in a galaxy of known distance
we find pulsating stars called Cepheids.
The period of pulsation of these Cepheid Stars is directly proportional
to their intrinsic luminosity, L. Once
we calibrate this relationship we find Cepheids in a more distant galaxy. Measuring the periods of pulsation we can
determine their intrinsic luminosity, measure their brightness and calculate
their distance:
-
--------------- D^2
= L cepheid / F * 4*pi
-
- CEPHEID EXAMPLE: 50 Cepheids are measured with pulsations
from 1 to 100 days. The brightness
measured and the luminosities calculated because the distance is known. Luminosities range from 1000 to 30,000 Solar
Luminosity. The straight line plot has
the equation: Luminosity =
320*(Period) + 575. To measure the distance to the M100 galaxy in the Coma Berenices three
Cepheid Stars are found to have periods of:
-
------- Periods -----Solar
------- Luminosity ------- Brightness----- Distance
------- Days ----- Luminosity------- watts -------- watts/meter----- MLY
-
------- 30 --------- 10,300-------
3.9*10^30 ----- 9.3*10^-19 -------- 61
------- 8 ---------
3,100--------- 1.2*10^30 -----
3.8*10^-19 -------- 53
------- 19--------- 10,300-------
3.9*10^30 ------ 8.7*10^-19 -------- 51
-
- Note that our calculations vary by 55 + or
- 5 million lightyears. The galaxy is
probably only a few hundred lightyears across so the error in our distance
measurement is + or - 10%. These are not
very accurate distance measurements.
-
- Besides the Cepheids in the Large
Megellanic Cloud galaxy another fortunate occurrence happened in 1987. A supernova exploded. Previous pulsations before the explosion had
created a shell of gas around the star. When the star exploded the ultraviolet
light took 240 days to reach the shell and light up the gas. The radius of the ring was measured to be
0.858 arc seconds.
-
- That distance would be the speed of light *
the 240 days = 6.2*10^15 meters. The sine of the angle 0.858 arc seconds =
4.12^10^-6. The distance to the star is
calculated to be 1.5*10^21 meters, which is 150,000 lightyears. This gives astronomers a better calibration
on the luminosity of the Cepheids and the luminosity of the supernova.
-
- OUT TO 10,000,000,000 LIGHTYEARS: Next we find a supernova explosion in a
galaxy still farther away. To know the
intrinsic luminosity it must be a certain type of supernova, called a white
dwarf supernova. Supernovae are much brighter than Cepheid stars and can be
measured at much greater distances.
-
- The peak luminosity of the supernova is
10^10 Solar Luminosity and it remains above 10^8 for 150 days. A typical Cepheid star is 10^4 Solar
Luminosity. We can measure the
brightness of Cepheids out to 100 million lightyears.
- Using the inverse distance squared formula
this means we can measure supernova out to 10^11 lightyears, 100 billion
lightyears. The Observable Universe is
10^10 or 10 billion lightyears.
Therefore, astronomers should be able to measure distances out to the
edge of the Universe.
-
- OUT TO 13,700,000,000 LIGHTYEARS: After measuring the distances to many
galaxies, Edwin Hubble noted the correlation of distance with recession
velocity. Recession velocities were
measured using the redshift of the galaxies light spectrum.
-
- Plotting distances of 200 to 1,600 million
lightyears versus receding velocities measured from 25,000 to 35,000 km/sec.
resulting in a straight line with a constant slope of 22,000 km/sec per million
lightyear. This is known as Hubble’s
Constant. It defines the rate of
expansion of the Universe. Galaxies will
be traveling faster by 22,000 km/sec, (49,000 miles per hour) for every million
lightyears distant they are away from us.
-
- HUBBLE EXAMPLE: We know the hydrogen emissions line in the
light spectrum is at 656.3 nanometers. We measure this in three different
galaxies and it has shifted to the red end of the spectrum. Redshift = z = (measured wavelength - 656.3) / 656.3
-
- For redshifts much less than 1 the
velocity = speed of light * z. The
receding velocity divided by 22,000 km/sec per million lightyears gives the
distance in million lightyears.
-
---------------Emission -----
Redshift ------- velocity ------------
Distance
----------- Wavelength
--------- z ----------- million mph-------- lightyears
-
------------659.6 nm ---------
.005 ----------- 3.4
------------------ 690,000
------------664.7 nm ---------
.013 ----------- 8.6
---------------- 1,750,000
------------6679.2 nm -------
.035 ----------- 23.4 ---------------- 4,760,000
-
- The last amazing thing about the Hubble
Constant for the expansion of the Universe is that lightyears are also time as
well as distance. And, if we run time
backwards we can determine when the expansion started.
-
- The reciprocal of the Hubble Constant will
give us the age of the Universe, assuming the expansion rate has always been
the same:
-
- 1/ 22,000 km/sec/ MLY =
1.36*10^10 years. The age of the
Universe is 13.6 billion years. And, the
distance to the edge of the Observable Universe is 13.6 billion lightyears.
-
May 15, 2023 GALACTIC
DISTANCES - measure them? 863 4006
----------------------------------------------------------------------------------------
----- Comments appreciated and Pass it on to
whomever is interested. ---
--- Some reviews are at: -------------- http://jdetrick.blogspot.com -----
-- email feedback, corrections, request for
copies or Index of all reviews
--- to: ------ jamesdetrick@comcast.net ------ “Jim Detrick” ---May 15, 2023
--------------------- --- Tuesday, May 16, 2023
---------------------------------
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