Exercise: Distance Ladder 5 - The relation between the brightness of SNe and galaxy’s distance
Version 2.0e

Exercise: The Relation between the Brightnessof Supernovae and the Distance to Galaxies
Guide for Teachers

Aim of this educational materials
The supernova explosion is used as a method to measure the distance to far galaxies. Because
the supernova explosion is very violent phenomenon, we can easily detect fairly far supernova’sappearance. Also, a part of supernova explosion which is called as Type Ia has a feature that theabsolute magnitude of maximum brightness becomes the same in any case. The absolutemagnitude is known in that type; therefore, by comparing with the apparent magnitude, we canget a clue about measuring the distance to the galaxy in which the supernova was found. This isa method to measure the distance to the cosmologically far galaxies.
In this exercise, 2 sets of materials are prepared. First, calculate some Type Ia supernova’s
absolute magnitudes and confirm that their values become almost constant. Second, on theassumption that the absolute magnitude of supernovae is already known, by measuring somesupernovae’s images, calculate distances to galaxies in which supernovae were found.

SN1994d visible

No supernova (by NASA, HST)

Preferable matters to be studied before these exercises
The preferable matters to be studied beforehand are fundamental knowledge about galaxies
and supernovae, relation between apparent magnitude and absolute magnitude and how to treat
logarithms.
The apparent magnitude of observed star m can be represented by comparison star’s
magnitude m0 and photometric value as follows.

m = m0 − 2.5 logll0l 0 : comparison star’s photometric valuel : observed star’s photometric value(c) PAOFITS Teaching Material [Guide for Teachers]

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Exercise: Distance Ladder 5 - The relation between the brightness of SNe and galaxy’s distance
Version 2.0e

This formula will be needed in the exercise 5-2.
The relation between absolute magnitude M, apparent magnitude m and distance r is asfollows.
R······· R = 10 pc
M = m + 5 log
rAccording to Richardson et al. (2002), the absolute magnitude of Type Ia supernovae’smaximum brightness is –19.46.
The distance to a galaxy is mainly calculated from radial velocity v by Hubble’s Law. As theHubble constant, we adopted the same value with Richardson et al. which was used when theycalculated the absolute magnitude.
v = H 0 × r ················· H 0 = 60 km/s/Mpc

Outline of the 2 sets of materials[Set 1] Research on the Relation between Type Ia Supernova’s Brightness and Galaxy’s
Distance - 1
Aim:
Find out that the value of Type Ia supernova’s absolute magnitude at maximum brightness
becomes around –19.5.
Exercise Plan (an example): 1 hour exercise in the classroom
1. About SN2002bo, from graph of the light curve, read the maximum brightness(magnitude).
2. Calculate the absolute magnitude from the result of 1.
3. By repeating the procedure of 1. and 2., calculate the absolute magnitude of eachsupernova.
4. Consider about the calculated values.
5. Calculate the distance for an object which distance is unknown. (If there is time.)
An example of the answer to the questions in worksheet:
Question 1 - 3(See Table 1 in next page.) When reading from the light curve, the variation of individual
student’s measurement doesn’t care. In this calculation, a table of logarithms is needed. It is easy
to execute this calculation by using calculator with a function to calculate logarithms or spread
sheet software.
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Exercise: Distance Ladder 5 - The relation between the brightness of SNe and galaxy’s distance
Version 2.0e

Because the number of supernovae is a lot, assign the work to the group composed of 3 or 4
students may be a good way.
Table 1
GalaxyNGC3190

Distance[Mpc]
26.2

Supernova
SN2002boApparent
magnitude
13.6

Calculated
magnitude-18.5

NGC3169

23.0

SN2003cg

14.4

-17.4

NGC546847.3

SN2002cr

14.2

-19.2

NGC767858.1

SN2002dp

15.0

-18.8

UGC10743

42.8

SN2002er

14.2

-19.0

NGC130935.6

SN2002fk

13.2

-19.6

NGC398775.8

SN2001V

13.8

-20.6

UGC11725

105.8

SN2001dl

16.2

-18.9

UGC10738112.0

SN2001cp

15.8

-19.4

UGC1162185.1

SN2001eh

16.4

-19.9

NGC144819.6

SN2001el

12.7

-18.8

UGC512967.0

SN2001fe

14.8-19.3

Mpc: represent one million parsec (pc). The parsec is a unit of distance defined as “adistance corresponding to a stellar parallax of one second”.

Question 4
Understand that absolute magnitudes concentrate around –19 magnitude is enough. Taking an
average will lead to question 5. An average from above table is –19.2.
Question 5
Calculate the distance from the absolute magnitude of question 4. The value around 75 Mpc
will be good enough. If you are planning to execute the exercise “Research on the Relation
between Type Ia Supernova’s Brightness and Galaxy’s Distance 2”, this question will beomitted.

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3Exercise: Distance Ladder 5 - The relation between the brightness of SNe and galaxy’s distance
Version 2.0e

[Set 2] Research on the Relation between Type Ia Supernova’s Brightness and Galaxy’s
Distance - 2
Aim:
Find out that the distance to a far galaxy can be measured by Type Ia supernova.
Class Plan (an example): 2 hours exercise in the computer room
1st hour
1. Specify the supernova and the comparison star in the image.
2. Execute photometry of the brightness of the comparison star and the supernova, andrecord them.
3. Calculate the apparent magnitude from photometric values.
2nd hour
1. Draw the light curve using the calculated apparent magnitude and read the maximum
brightness (magnitude).
2. Calculate the distance from the formula for the apparent magnitude and the absolutemagnitude at maximum brightness.
3. Consider about calculated values.

An example of the answer to the questions in worksheet:
Question 1(See Figure 1.) A supernova exists in thedimmed galaxy at center. By adjusting thebrightness of the image using the triangle
mark of the slide bar, the galaxy will appear
more clearly.
Question 2
Either of two comparison stars can be used.
Average of both stars will lead more accuratevalue.
Question 3
Figure 1
Calculation of logarithms is difficult. Useof spread sheet software is a good way. Please use the prepared Excel file.

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Exercise: Distance Ladder 5 - The relation between the brightness of SNe and galaxy’s distance
Version 2.0e

Question 4
When drawing a light curve, light curves of supernovae in exercise 5-1 can be referred. Themagnitude at maximum brightness around 13.2 will be good.
Question 5
The value around 30 –35 [Mpc] will be good.
Question 6
This value was calculated from Hubble’s Law with the radial velocity 1921[km/s] of thegalaxy UGC9391 in which SN2003du was observed.

NOTE: The maximum brightness of supernovae
Themaximum brightnessofType

Ia

supernovae is stable compared with those of othertypes, but still, show somewhat individuality asFigure 2 (upper panel). To determine themaximum brightness strictly, read the maximum
brightness applying the curve fit method at thedescending part of the light curve. That’s because
the maximum brightness will change according to
the descending speed of light. In general, slowly
descending supernova’s maximum brightness
becomes more brightened. Considering this fact,
the maximum brightness becomes almost –19.5 as
Figure 2 (under panel). But, without suchcorrections, the difference of the maximum
magnitude between bright supernova and darksupernova becomes 1 magnitude. This is one of
the reasons why student’s value doesn’t become–19.5 clearly. In this exercise, the individualitywill become decrease due to a lot of data and theresult will become good value. Therefore, pleasecalculate average without much attention.
One more reason why student’s value doesn’t
approach –19.5 is that the student tends to read themaximum observed point when reading the(reference; http://www-supernova.lbl.gov/)
Figure 2.

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Exercise: Distance Ladder 5 - The relation between the brightness of SNe and galaxy’s distance
Version 2.0e
maximum brightness. Guidance of procedure that imagine as drawing a smoothed line first, and then read themaximum value of the line, will lead a good result. As another way, without speaking such procedure, makestudents consider about values later.

Source of materials:* Research on the relation between Type Ia supernova’s brightness and galaxy’s distance – 1
Light curves of supernovae:
Supernovae Home Page
http://www.astrosurf.com/snweb/
Radial velocities of galaxies:
NAOJ/ADAC, On-line Catalog Service, Galaxy Redshift Catalog
Images of galaxies:
NAOJ/ADAC, HST archive
Absolute magnitude:
Dean Richardson et al., The Astronomical Journal,123:745-752,2002,Feb* Research on the relation between Type Ia supernova’s brightness and galaxy’s distance - 2
Radial velocities (distance):
NAOJ/ADAC, On-line Catalog Service, Galaxy Redshift Catalog
Images of galaxies:
NAOJ/ADAC, HST archive
Images of supernovae:
Courtesy of Mr. Yasuo Sano, Kihara Observatory, Nayoro city

Supplement
Corresponding to the National Curriculum Guidelines in Japan:
Senior High School
Earth Science & Astronomy I(2) Composition of Atmosphere, Ocean and Universe
Composition of Universe, Galaxy and Universe
Earth Science & Astronomy II(4) Research of Universe, Extension of Universe,Distance and Mass of Celestial Object, Composition of Universe(c) PAOFITS Teaching Material [Guide for Teachers]

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