1. The Distance to the Stars
A Most Difficult Problem
How Might You Do This?
If you knew the size you could measure the subtended angle and determine the distance
You do not know the size.
Stars appear as geometric points (except for 1 or 2 cases).
If you knew the absolute amount of energy given off and the amount received (per m2) at the Earth then you could calculate the distance.
We do not know the energy given off.
Very difficult measurement.
What if there is intervening material which is scattering the energy?

2. Parallax? Apparent Change in Angle Due to Motion of the Observer
Generally Denoted π
Apparent Change in Angle Due to Motion of the Observer

3. What Do We Need to Know First?
So Can This Be Measured?
What Do We Need to Know First?
Is the Earth moving about the Sun?
Aberration of Starlight.
Change in Velocity of Stars During the Course of a year.
What is the Distance From the Earth to Sun?
1 AU = 1.49(108) km

4. Stellar Aberration
First noticed in 1729 by James Bradley while trying to measure parallax.
Assumed that parallax would move nearby stars during transit to different places relative to a fixed point (or line).
All stars were observed to move by 20.5” (amplitude) during the course of a year!
What matters is the ratio of the forward speed to the speed of the incoming particle.

5. Stellar Aberration The total amount of time is t d α dN
In t the Earth (Cyan Vector) moves vt = dN
In t light (Orange) moves ct = d
vt/ct = dN/d = tan(α)
The measured angle is 20.5" (maximum which corresponds to the Earth moving at right angles to the object).
v = tan(α) * c = 29 km/s d α dN

6. The Effect on Position

7. Radial Velocity Changes
What is a Radial Velocity?
Vr = Line of Sight Velocity
V is the space velocity
T is the tangential velocity
Note that the plane for the velocities is defined by Vr and V and that T must be  to Vr
A job for the Doppler Effect

8. Take Two Spectra Six Months Apart
To Maximize the Effect Make Sure the Star is in the Plane of the Ecliptic
Arcturus on 1 July 1939 and on 19 January 1940
RV = +18 km/s on 1 July and -32 km/s on 19 January
The difference (50 km/s) is due to the orbital motion of the Earth.

9. Parallax
First parallax (61 Cygni) was measured in 1838 by Bessel

10. The Definition of Parallax
Parallax (π) is defined as the maximum angle subtended by the Earth’s semimajor axis. The parallax is the semimajor axis of the parallatic ellipse.
1 AU = dπ where π is in radians.
The usual unit for π is arcseconds: 1 radian = seconds
1 AU = dπ/206265
So if π = 1 arcsecond d = AU = 1 parsec.
d (in parsecs) = 1/π

11. Parallax Measurements
The largest parallax is α Cen C (Proxima) at π = 0.763" (1.31 pc) followed by α Cen at 0.741" (1.35 pc)
The smallest ground based measure is about 0.01" (100 pc) but the precision is  0.01".
Hipparchus went to about 0.005" (200 pc) with an uncertainty of about  0.005".
This is a rather broad generalization as the uncertainty depends not only upon the size of the parallax but the apparent brightness of the star and the crowding of the field.
Hipparchus measured about 250 stars within 15 parsecs and more than within 100 parsecs.

12. Other Distance Methods
All Are Indirect!
Stellar Motions
The closer you are the faster you appear to move.
Moving Clusters
Moving towards a convergent point - common motion and velocity will yield a distance.
Inverse Square Law
Need to know intrinsic brightness and line of sight extinction.
Dynamical Parallaxes
Binaries can yield physical separation in real units.
Interstellar Lines
Strength is proportional to distance.
Galactic Rotation
Distant stars in the disk have speeds which are dominated by galactic rotation.

13. Magnitudes Brightest Stars: m ~ 0 (apparent!)
Historic System
Brightest Stars: m ~ 0 (apparent!)
Faintest Naked Eye Stars: m ~ 6
Visual Magnitudes: Defined by the sensitivity of the eye
Photographic Magnitudes: “Blue” photographic emulsion (esp IIaO)
V : mV defined by the V filter of the Johnson UBV system and is often called the visual magnitude.

14. The Definition of Magnitudes
Pogson’s Ratio
A difference of five (5) magnitudes is defined as an energy ratio of 100.
Therefore: 1 magnitude = 100^0.2 = difference in energy.
= 100
If star 1 is 1 magnitude brighter than star 2 then:
where l is the received energy (ergs, photons). l / l = 2 .
512 1 2

15. More on Magnitudes
In general: If star 1 is of magnitude m1 and star 2 is of magnitude m2 (star 1 brighter than star 2):
The - sign is necessary as the brighter the star the numerically less the magnitude.
This means: m1 = -2.5 log l1 and m2 = -2.5 log l2!
Now convert the equation to base 10:

16. Apparent and Absolute Magnitudes
Apparent Magnitude:
Magnitude as observed on the Earth
Sirius has apparent magnitudes of
mV = -1.46
mB = -1.46
mU = -1.51
Apparent magnitudes depend on wavelength.
Absolute Magnitude
(Apparent) Magnitude a star would have if it were at a distance of 10 parsecs from the Sun
Makes it possible to directly compare intrinsic brightnesses of stars.
Just like apparent magnitudes absolute magnitudes are wavelength dependent.

17. Absolute Magnitudes Let ld = energy observed from star at distance d
Let l10 = energy observed from star at 10 pc
Call md the apparent magnitude m and m10 the absolute magnitude M.
d = distance in parsecs.

18. The Distance Modulus

19. Distance Modulus II m - M is the distance modulus
0 = 10 pc
5 = 100 pc
Each 5 magnitude increase is a factor of 10 in distance: 102 = 100 ==> 5 magnitudes!
If there is interstellar absorption then
5 log (d/10) = m - M - A
A = absorption in magnitudes.
m = apparent (observed) magnitude
m = (mo + A) where mo is what the observed magnitude would be if there were no absorption.
m, M, and A are all wavelength dependent.

20. Distances to Stars
Example 1: What would the apparent magnitude of a star with MV = +5 be at a distance of 10 pc?
m - M = 5 log (d/10) so m = log (10/10) so m = 5.
Note that this is the standard distance for absolute magnitudes and that the Sun’s absolute magnitude is about +5. The Sun is a dwarf or Main Sequence star.
m = 5 is at the lower end of naked eye stars.
Example 2: What would be the distance to a star with absolute magnitude -5 and apparent magnitude +5?
m - M = 10 = 5 log (d/10) ==> d = 1000 pc
This is a evolved supergiant star or an unevolved early type dwarf.
Note that if you put this star at 10 pc its apparent magnitude would be 5 which is brighter than Venus at maximum.
The closest star of this type is several hundred parsecs away.

21. Bolometric Magnitude
Bolometric magnitude is the magnitude that corresponds to the total luminosity. Remember that the definition of magnitudes does not include a zero-point!
It can be either an apparent or absolute magnitude.

22. Solar Bolometric Magnitude
What is the bolometric magnitude of the Sun?
mv = BC = -0.07
mv - Mv = 5 log (d/10)
d = (206265)-1 pc
Plug in the numbers: MV = and Mbol = +4.76
FYI: The solar luminosity = 3.82 (1033) ergs s-1

23. Color Indices and Interstellar Extinction
λi; A(λi); Apparent Magnitude mio
λj; A(λj); Apparent Magnitude mjo
mio and mjo are what you would get if A = 0
Observed Apparent magnitudes:
mi = mio + A(λi)
mj = mjo + A(λj)
Color Index Cij = mi - mj (observed) Cijo = mio – mjo (intinsic)

24. Color Excess For UBV: E(B-V) = (B-V) - (B-V)o E(U-B) = (U-B) - (U-B)o
E(B-V) and E(U-B) are positive numbers and mean that the observed colors are redder than the intrinsic colors.
A more positive B-V is redder as V (the redder magnitude) is brighter (intrinsically smaller) than B.

25. Computing a Magnitude But what is CB?
1) Assume that all Johnson magnitudes for Vega are 0
2) Assume that the temperature of Vega is 9400K and that
the energy distribution of Vega is a blackbody then CB is just
-2.5 log(lB) using E = Planck function and T = 9400K. CB
is the correction to Johnson B for all temperatures.
3) For UVRI proceed in the same fashion. 25