1. OPTION E - ASTROPHYSICS E3 Stellar distances Parallax method

2. Astronomical distances – recap
The SI unit for length, the metre, is a very small unit to measure astronomical distances. There units usually used is astronomy:
The Astronomical Unit (AU) – this is the average distance between the Earth and the Sun. This unit is more used within the Solar System.
1 AU = km or
1 AU = 1.5x1011m

3. Astronomical distances – recap
The light year (ly) – this is the distance travelled by the light in one year.
c = 3x108 m/s
t = 1 year = x 24 x 60 x 60= 3.16 x 107 s
Speed =Distance / Time
Distance = Speed x Time
= 3x108 x 3.16 x 107 = 9.46 x 1015 m
1 ly = 9.46x1015 m

4. E3 - Stellar distances
E.3.1 Define the parsec.
The parsec (pc) – this is the distance at which 1 AU subtends an angle of 1 arcsencond.
“Parsec” is short for
parallax arcsecond
1 pc = 3.086x1016 m or
1 pc = 3.26 ly

5. (206,000 times further than the Earth is from the Sun)
E3 - Stellar distances
1 parsec = X 1016 metres
Nearest Star
1.3 pc
(206,000 times further than the Earth is from the Sun)

6. E3 - Stellar distances
E.3.2 Describe the stellar parallax method of determining the distance to a star.
Bjork’s Eyes
Space
Where star/ball appears relative to background
Angle star/ball appears to shift
Distance to star/ball
“Baseline”

7. E.3.2 Describe the stellar parallax method of determining the distance to a star.
Parallax, more accurately motion parallax, is the change of angular position of two observations of a single object relative to each other as seen by an observer, caused by the motion of the observer.
Simply put, it is the apparent shift of an object against the background that is caused by a change in the observer's position over a period of 6 months.

8. E3 - Stellar distances
E.3.2 Describe the stellar parallax method of determining the distance to a star.
We know how big the Earth’s orbit is, we measure the shift (parallax), and then we get the distance…
Parallax - p
(Angle)
Distance to Star - d
Baseline – R
(Earth’s orbit)

9. E3 - Stellar distances
E.3.2 Describe the stellar parallax method of determining the distance to a star.
For very small angles tan p ≈ p
In conventional units it means that

10. E3 - Stellar distances
E.3.2 Describe the stellar parallax method of determining the distance to a star.

11. Angular sizes 360 degrees (360o) in a circle
E3 - Stellar distances
Angular sizes
360 degrees (360o) in a circle
60 arcminutes (60’) in a degree
60 arcseconds (60”) in an arcminute

12. E3 - Stellar distances
E.3.3 Explain why the method of stellar parallax is limited to measuring stellar distances less than several hundred parsecs.
The farther away an object gets, the smaller its shift.
Eventually, the shift is too small to see.
Measurements from Earth only allow distances up to 300ly, or roughly 100pc, to be determined with the parallax method.
With satellites, distances of around 500 pc can be determined.

13. How Do We Measure the Distance to Stars? - Instant Egghead #46

14. Go to Option E – Astrophysics SL worksheet
E.3.4 Solve problems involving stellar parallax.
Go to Option E – Astrophysics SL worksheet

15. OPTION E - ASTROPHYSICS E3 Stellar distances
OPTION E - ASTROPHYSICS E3 Stellar distances Absolute and apparent magnitudes

16. Another thing we can figure out about stars is their colours…
E3 - Stellar distances
E.3.5 Describe the apparent magnitude scale.
Another thing we can figure out about stars is their colours…
We’ve figured out brightness, but stars don’t put out an equal amount of all light…
…some put out more blue light, while others put out more red light!

17. E3 - Stellar distances
E.3.5 Describe the apparent magnitude scale.
Usually, what we know is how bright the star looks to us here on Earth…
We call this its Apparent Magnitude
“What you see is what you get…”

18. Betelgeuse and Rigel, stars in Orion with apparent magnitudes
E3 - Stellar distances
E.3.5 Describe the apparent magnitude scale.
Magnitudes are a way of assigning a number to a star so we know how bright it is
Similar to how the Richter scale assigns a number to the strength of an earthquake
Betelgeuse and Rigel, stars in Orion with apparent magnitudes
0.3 and 0.9
This is the “8.9” earthquake off of Sumatra

19. The historical magnitude scale…
E3 - Stellar distances
The historical magnitude scale…
Greeks ordered the stars in the sky from brightest to faintest…
…so brighter stars have smaller magnitudes.
Magnitude
Description
1st
The 20 brightest stars
2nd
stars less bright than the 20 brightest
3rd
and so on...
4th
getting dimmer each time
5th
and more in each group, until
6th
the dimmest stars (depending on your eyesight)

20. Later, astronomers quantified this system.
E3 - Stellar distances
Later, astronomers quantified this system.
Because stars have such a wide range in brightness, magnitudes are on a “log scale”
Every one magnitude corresponds to a factor of 2.5 change in brightness
Every 5 magnitudes is a factor of 100 change in brightness
(because (2.5)5 = 2.5 x 2.5 x 2.5 x 2.5 x 2.5 = 100)

21. Brighter = Smaller magnitudes Fainter = Bigger magnitudes
E3 - Stellar distances
Brighter = Smaller magnitudes Fainter = Bigger magnitudes
Magnitudes can even be negative for really bright objects!
Object
Apparent Magnitude
The Sun
-26.8
Full Moon
-12.6
Venus (at brightest)
-4.4
Sirius (brightest star)
-1.5
Faintest naked eye stars
6 to 7
Faintest star visible from Earth telescopes
~25

22. 𝑏 𝑏 0 = 2.512 −𝑚 E.3.5 Describe the apparent magnitude scale.
E3 - Stellar distances
E.3.5 Describe the apparent magnitude scale.
Given a star of apparent brightness b, we assign that start and apparent magnitude m defined by:
𝑏 𝑏 0 = 100 −𝑚 5
Where 𝑏 0 =2.52× 10 −8 𝑊 𝑚 −2 is taken as the reference value for apparent brightness
Or:
𝑚=− 5 2 𝑙𝑜𝑔 𝑏 𝑏 0
The first equation can also be re-written as:
𝑏 𝑏 0 = −𝑚

23. E3 - Stellar distances
E.3.6 Define absolute magnitude.
However: knowing how bright a star looks doesn’t really tell us anything about the star itself!
We’d really like to know things that are intrinsic properties of the star like: Luminosity (energy output) and Temperature

24. …we need to know its distance!
E3 - Stellar distances
E.3.6 Define absolute magnitude.
In order to get from how bright something looks…
to how much energy it’s putting out…
…we need to know its distance!

25. Absolute Magnitude (M):
E3 - Stellar distances
E.3.6 Define absolute magnitude.
The whole point of knowing the distance using the parallax method is to figure out luminosity…
Once we have both brightness and distance, we can do that!
It is often helpful to put luminosity on the magnitude scale…
Absolute Magnitude (M):
The magnitude an object would have if we put it 10 parsecs away from Earth

26. Absolute Magnitude (M)
E3 - Stellar distances
E.3.6 Define absolute magnitude.
Absolute Magnitude (M)
removes the effect of distance and puts stars on a common scale
The Sun is in apparent magnitude, but would be 4.4 if we moved it far away
Aldebaran is farther than 10pc, so it’s absolute magnitude is brighter than its apparent magnitude
Remember magnitude scale is “backwards”

27. Absolute Magnitude (M)
E3 - Stellar distances
Absolute Magnitude (M)
Knowing the apparent magnitude (m) and the distance in pc (d) of a star its absolute magnitude (M) can be found using the following equation:
Example: Find the absolute magnitude of the Sun.
The apparent magnitude is -26.7
The distance of the Sun from the Earth is 1 AU = 4.9x10-6 pc
Therefore, M= – log (4.9x10-6) + 5 =
= +4.8

28. So we have three ways of talking about brightness:
E3 - Stellar distances
So we have three ways of talking about brightness:
Apparent Magnitude - How bright a star looks from Earth
Luminosity - How much energy a star puts out per second
Absolute Magnitude - How bright a star would look if it was 10 parsecs away

29. E.3.7 Solve problems involving apparent magnitude, absolute magnitude and distance.
E.3.8 Solve problems involving apparent brightness and apparent magnitude.

30. OPTION E - ASTROPHYSICS E3 Stellar distances Spectroscopic parallax

31. E.3.9 State that the luminosity of a star may be estimated from its spectrum.
Spectroscopic parallax is an astronomical method for measuring the distances to stars.
Despite its name, it does not rely on the apparent change in the position of the star.
This technique can be applied to any main sequence star for which a spectrum can be recorded.

32. E.3.10 Explain how stellar distance may be determined using apparent brightness and luminosity.
The Luminosity of a star can be found using an absorption spectrum.
Using its spectrum a star can be placed in a spectral class.
Also the star’s surface temperature can determined from its spectrum (Wien’s law)
Using the H-R diagram and knowing both temperature and spectral class of the star, its luminosity can be found.

33. E.3.11 State that the method of spectroscopic parallax is limited to measuring stellar distances less than about 10 Mpc.
E.3.12 Solve problems involving stellar distances, apparent brightness and luminosity.

34. Distance measurement by parallax apparent brightness spectrum
Distance measured by parallax:
Distance measurement by parallax
apparent brightness
spectrum
Chemical composition of corona
Wien’s Law (surface temperature T)
Luminosity
L = 4πd2 b
d = 1 / p
L = 4πR2 σT4
Stefan-Boltzmann
Radius

35. OPTION E - ASTROPHYSICS E3 Stellar distances Cepheid variables

36. E.3.13 Outline the nature of a Cepheid variable.
Cepheid variables
Cepheid variables are stars of variable luminosity.
The luminosity increases sharply and falls of gently with a well-defined period.
The period is related to the absolute luminosity of the star and so can be used to estimate the distance to the star.
A Cepheid is usually a giant yellow star, pulsing regularly by expanding and contracting, resulting in a regular oscillation of its luminosity.
The luminosity of Cepheid stars range from 103 to 104 times that of the Sun.

37. E.3.14 State the relationship between period and absolute magnitude for Cepheid variables.
The relationship between a Cepheid variable's luminosity and variability period is quite precise, and has been used as a standard candle (astronomical object that has a known luminosity) for almost a century.
This connection was discovered in 1912 by Henrietta Swan Leavitt. She measured the brightness of hundreds of Cepheid variables and discovered a distinct period-luminosity relationship.

39. E.3.14 State the relationship between period and absolute magnitude for Cepheid variables.
A three-day period Cepheid has a luminosity of about 800 times that of the Sun.
A thirty-day period Cepheid is 10,000 times as bright as the Sun.
The scale has been calibrated using nearby Cepheid stars, for which the distance was already known.
This high luminosity, and the precision with which their distance can be estimated, makes Cepheid stars the ideal standard candle to measure the distance of clusters and external galaxies.

40. E.3.14 State the relationship between period and absolute magnitude for Cepheid variables.

41. E.3.15 Explain how Cepheid variables may be used as “standard candles”.
The luminosity of a Cepheid variable can be determined from its period.
The brightness of the Cepheid (b) can be determined from its apparent magnitude.
Then, from the relationship
𝒃= 𝑳 𝟒𝝅 𝒅 𝟐
the distance to the Cepheid can be determined.
If a Cepheid variable is located in a particular galaxy, then the distance to the galaxy may be determined.
The Cepheids method can be used to find distances up to a few Mpc.

42. Surface temperature (T)
E.3.16 Determine the distance to a Cepheid variable using the luminosity–period relationship.
Distance measured by spectroscopic parallax / Cepheid variables:
Apparent brightness
Luminosity class
spectrum
Chemical composition
Cepheid variable
Spectral type
H-R diagram
Surface temperature (T)
Wien’s Law
Period
Luminosity (L)
Stefan-Boltzmann
L = 4πR2 σT4
b = L / 4πd2
Distance (d)
Radius