### Lecture 3

```The Sun and the Stars
The Sun and the Stars
Dr Matt Burleigh
The Sun and the Stars
Stellar distances
An accurate distance scale is a fundamental tool in Astronomy. Without accurate distances we can not determine
fundamental stellar properties, for example, stellar radii, luminosities etc.
But how do we measure stellar distances?
Geometric Parallax (g)
Geometric parallax (g ) is the angular displacement of a relatively nearby object with respect to distant
(assumed to be at infinity) background objects.
Easiest to demonstrate by observing your thumb at arms length first with your left eye and then with your right eye.
You should see your thumb displaced relative to the background objects.
e.g. consider the following geometry

 d tan
2
d 
g
For large distances, i.e. small parallaxes, we can use
the small angle approximation, i.e.
2

2 tan
g
tan
2
g

2
and so
d 
g
2

g
Dr Matt Burleigh
The Sun and the Stars
Stellar distances : Part 1 The Earth-Sun distance
1) Solar parallax s
For the sun, the solar parallax s is defined to be the angle subtended by the Earth’s radius at the distance of the sun
Re
 s
ds
Re
Thus, if we know the Earth’s radius, we can measure the
distance to the sun, by measuring the solar parallax
ds
However, the solar parallax is small (~8.8”, or the size of a
small coin at ~200 hundred meters). Also during the day
difficult to see background stars
Dr Matt Burleigh
The Sun and the Stars
2) Aristarchus of Samos (c310-230 BC)
Made the first measurement of the relative distance Earth-Moon : Earth-Sun. Though the result was inaccurate,
demonstrated that the Sun was very distant compared with the moon.
Relative distance
Measure the angle  between Moon’s terminator and Sun, when
Moon is half-illuminated.
Then,
dE
 M
dE
 S
 cos 
Since the Sun is very distant,  is close to 90 degrees.
Aristarchus measured an angle of 87 degrees, placing
the Sun approx. 20 times further away than the Moon.
[The correct scaling is ~390]. Error most likely associated with
determining precisely when Moon is half-illuminated.
Dr Matt Burleigh
The Sun and the Stars
3) Alternative approach : measure the parallax to another body in the solar system, e.g. a planet, and use it’s distance
and Keplers III law (P2  a3) to derive the Earth-Sun distance.
4) transit of Venus (Halley 1716)
Edmund Halley in 1716 suggested that the transit of Venus could be used to provide an accurate estimate of the
Earth-Sun distance.
Transits occur when the planet passes between the Earth and Sun. NB. Transits of Venus are rare, because the
orbit of Venus is inclined to the ecliptic (the plane of the Earth’s orbit) by 3.394 degrees.
As Venus passes between Earth and Sun, Venus occults (blocks) the light from the solar disc, appearing as a
When observed from 2 different locations on the Earth, the direction of the “shadow” is displaced due to parallax.
Dr Matt Burleigh
The Sun and the Stars
Last Transits of Venus were 2004 and 2012. Next transit is in 2133!
Dr Matt Burleigh
The Sun and the Stars
Stellar distances : Part 2 The stars
Trigonometrical Parallax :
as the Earth orbits the sun, a nearby star will appear to move
relative to a set of background stars.
The parallax ,p, of a star is defined to be half the total
angular change in direction to the star. At a distance of
1 parsec (par for parallax, sec for second), 1AU subtends
1 second of arc.
d(parsecs) = 1/p(arcseconds) = 1/”
1 parsec = 3x1018cm =206,265 AU = 3.262 light-years
The parallax of our nearest star alpha-Centauri is 0.76”, so
d(-Cen)=1/0.76=1.3pc=4.2 light-years.
The Hipparchus satellite could measure angular separations of
0.002”
(i.e. distances to few thousand stars)
ESA’s Gaia mission (Launched just before Xmas 2013) will measure
angular separations to a few -arc seconds
-6
(10 arc seconds), i.e. distances to a billion stars….!!!
Dr Matt Burleigh
The Sun and the Stars
Proper motion
Because Sun and stars move through space, nearby stars slowly drift in position across the sky.
This annual motion is called the proper motion () and is measured in seconds of arc/yr.
C
e.g. If star moves from B to D in 1 yr, then angular
movement observed from sun at A is . Distance d,
to the star at the end of year is AD.
In triangle ACD,
sin m =
If  is small,
If the annual parallax
 "
1
d
then
V trans  4 . 74
"
"
m=
Vtrans
d
Vtrans
d
measure  in arcsecs/yr
d in parsecs
and Vtrans in au/yr
[1au/1yr = 150,000,000/(365x24x3600) = 4.74 km/s]
The largest proper motion measured is that of Barnard’s star ~10.3”/yr
Dr Matt Burleigh
The Sun and the Stars
Motion of Barnard’s star on celestial sphere
is a combination of the proper motion of the
Star and the parallax due to Earth’s motion
around the Sun.
Dr Matt Burleigh
The Sun and the Stars
.
Apparent magnitude : The apparent magnitude (symbol m) is a measure of the stars brightness
as seen by an observer on Earth. Scale originally devised by Hipparchus and later Ptolemy.
Historically , stars were divided into 6 categories according to their brightness :
brightest 1st magnitude, faintest 6th magnitude.
Pogson (1836) verified Herschel’s finding that a difference of 5 magnitudes is equivalent to a factor
of 100 in received flux. But a difference of one magnitude is equivalent to a factor ~2.5.
Q. Why?
A. the eye estimates differences in brightness by a geometric rather than an arithmetic progression. Ie the eye’s
brightness scale is logarithmic rather than linear.
Pogson devised a logarithmic scale relating the apparent brightness of 2 stars (m1 and m2) in terms
of their measured fluxes, f1 and f2, such that
æf ö
m1 - m2 = -2.5log10 ç 1 ÷
è f 2ø
NB when m1 > m2, f1 < f2, ie brighter stars have smaller apparent magnitudes!!
Dr Matt Burleigh
The Sun and the Stars
Distance modulus:
Luminous stars can appear dim if very far away, while low-luminosity stars can appear bright
if very close.
f 
L
d
2
In Astronomy we use the absolute magnitude (symbol M) to refer to a stars luminosity
The absolute magnitude is defined to be the magnitude that would be observed if the star
were placed at a distance of 10 parsec from the sun.
m and M are related by the distance modulus m – M, where
 d 
m  M  5 log 

 10 
m  M  5 log d  5
Follows from:
 f
m 1  m 2   2 . 5 log 
 f
2
 d 
 d 
  

f
D
 10 
L
1
2




2
M  m  5  5 log d
M  m  5  5 log  "
Dr Matt Burleigh
The Sun and the Stars
Magnitude systems
Because the flux of a star varies with wavelength, the star’s magnitude will depend upon both
the wavelength at which we observe and the filter band-pass.
Historically we have used photographic mpg (4200A) and visual magnitudes mv (5500A).
The design of the filters is of paramount importance when devising photometric magnitude systems.
e.g. small band-passes give more spectral information but admit less flux requiring longer exposure times
Dr Matt Burleigh
The Sun and the Stars
Johnson – devised first photometric system using 3 filters U, B ,V
(Ultraviolet, Blue and Visible), centred at 3600, 4400, 5500A respectively.
To calibrate the system require measurement of flux standard stars for a
given instrument and filter set. Pioneering work by Alan Cousins in South Africa
Dr Matt Burleigh
The Sun and the Stars
Colour Index
The colour index CI is defined to be the difference between magnitudes at two
different effective wavelengths
CI = mB - mv = M B - M v
or, more generally
CI = m( l1 ) - m( l2 )
NB. U-B and B-V are also colour indices
Dr Matt Burleigh
The Sun and the Stars
Hotter stars have more negative colour indices!!
Dr Matt Burleigh
The Sun and the Stars
Extinction
Measured fluxes must be corrected for 2 types of extinction
(i) Atmospheric (from ground only – obviously!)
(ii) Interstellar
Zenith
Target
(i) Atmospheric – the attenuation due to the Earth’s
atmosphere depends upon the altitude of the star.
If F0() is the stars flux above the Earths atmosphere,
and  is the atmospheric optical depth, then the
observed flux F() is simply
F(l ) = F0 (l )e-t ( l )
z
altitude
The optical depth is a minimum at the zenith, 0(),
and proportional to sec Z, where Z is the zenith
angle, at other elevations (see fig), that is
Horizon
 (  )   0 (  ) sec Z
So if we know 0() we can measure the flux outside the atmosphere
Dr Matt Burleigh
The Sun and the Stars
The extinction for a given night can be determined by measuring standard stars through a range
of elevations (range of air masses) – derive extinction in magnitudes per unit air mass
m(l )- m0 (l ) = -2.5log(e-t ( l ) )
m 0 (  )  m (  )  1 . 086  (  )
Substitute for ()
m 0 (  )  m (  )  1 . 086  0 (  ) sec Z
1.0860() is the first order extinction coefficient
(ii) Interstellar extinction – e.g. scattering by gas and/or dust
 m (  )  1 . 086  (  )  A (  )
A() difficult to determine, varies with wavelength and line of sight through ISM.
Modifies distance modulus accordingly
m  M  5 log d  5  A
Dr Matt Burleigh
The Sun and the Stars
Blue photons scattered more than red, so stars are observed to have a colour excess, i.e.
they will appear redder. If a star has an intrinsic colour B0-V0, , and apparent B and V magnitudes
V  M V  5 log d  5  AV
and
B V  M
B
B  M
B
 5 log d  5  A B
then
 M V  A B  AV
The stars intrinsic colour B0-V0 =MB-MV
The colour excess E(B-V) is defined to be
E (B  V )  (B  V )  (B0  V0 )
Rearranging, we get
B  V  (B0  V0 )  E (B  V )
Typically we find that AV3E(B-V)
Dr Matt Burleigh
The Sun and the Stars
Bolometric Correction
In general, we only sample a small fraction of the stars energy distribution.
Above the Earth’s atmosphere we can define the total bolometric flux fbol ,where

f bol 

f ( )d 
0
It follows that
m bol   2 . 5 log f bol  const
Similarly we can define the visual flux, fv
The bolometric correction BC is defined
to be the difference between the
bolometric and visual magnitudes
BC  m bol  m v  M bol  M v

fv 

f ( ) S ( )d 
0
where S() is our sensitivity at V, and so
 f
BC   2 . 5 log  bol
 fv




This implies
m v   2 . 5 log f v  const
M bol  m v  5  5 log d  A v  BC
Dr Matt Burleigh
```