Download Module1: Scale of the Universe

Module 1
Scale of the Universe
The Cosmic Distance Ladder:
Parallax
Activity Guide
PARALLAX: ACTIVITY GUIDE
1
Introduction
Determining the position of a star or other object in space is an important concept in
astronomy. During this activity you will learn how the distances to nearby stars can be
measured using the parallax effect, and put this method into practise to determine the
distance to nearby stars.
Materials List
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LCOGT&Galaxy&image&
LCOGT&Star&image&
Student&worksheet&
Chalk&
Measuring&wheel&
Astrolabe&
Background Science
When&looking&at&the&night&sky,&it’s&fairly&easy&to&measure&how&the&stars&differ&in&
brightness,&but&discerning&how&distant&the&stars&are&is&a&more&difBicult&task.&One&way&to&
measure&astronomical&distances&is&called&Parallax.&&
Parallax&is&the&apparent&change&in&position&of&a&nearby&object&caused&by&a&change&in&the&
observer’s&point&of&view.&&This&is&demonstrated&in&the&diagram&below;&as&the&observer&
moved&between&two&positions,&he&would&see&the&same&tree&but&it&would&appear&to&move&
against&the&background.&
To&calculate&distances&in&space,&astronomers&measure&the&parallax&angle&as&
demonstrated&on&the&diagram&below:&
Once&the&parallax&angle&has&been&measured&twice,&from&opposite&sides&of&Earth’s&orbit&
around&the&Sun,&we&can&calculate&the&distance&to&stars&with&the&parallax&method&we&use&
the&following&equation:&
PARALLAX: ACTIVITY GUIDE
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&
The&parallax&angle&is&measure&in&arcminutes&(arcmin)&and&arcseconds&(arcsec&.&Just&as&
an&hour&is&divided&into&60&minutes&and&a&minute&into&60&seconds,&a&degree&is&divided&
into&60&arcminutes&and&an&arcminute&is&divided&into&60&arcseconds.&&
1&degree&=&1°&=&1/360&of&a&circle&
1&arcminute&=&1'&=&1/60&of&a&degree&
1&arcsecond&=&1"&=&1/60&of&an&arcminute&=&1/3600&of&a&degree&
The&parallax&method&can&only&be&used&to&measure&the&distance&to&stars&close&enough&to&
show&a&measurable&parallax,&this&stretches&to&about&100&parsecs&(or&~326&light&years).&
Instructions
In&this&activity&you&will&investigate&how&the&distance&of&an&object&is&related&to&how&far&it&
appears&to&move&when&you&view&it&from&different&perspectives&and&discuss&how&
parallax&can&be&used&in&astronomy&to&reveal&the&distance&to&nearby&stars.&
i.#Preparation#
Before&the&lesson,&you&will&need&to&locate&a&suitable&area&to&create&your&parallax&
diagram&such&as&a&playground&that&will&not&be&disturbed.&Ensure&you&have&all&the&
materials&necessary&(see&Materials+List)&and&follow&the&steps&below:&
1.& Draw&a&straight&line&24^metres&long&with&chalk.&Mark&the&centre&of&the&line&with&
a&circle.&This&will&represent&the&Sun.&&
2.& Measuring&along&the&line&to&one&side&of&the&Sun,&draw&circles&at&1^metre,&6^
metres&and&10^metres.&These&circles&will&represent&Planet&1,&Planet&3&and&Planet&
5&consecutively.&&
3.& To&the&other&side&of&the&Sun,&draw&circles&at&5^metres,&8^metres&and&12^metres.&
Label&these&as&Planet&2,&Planet&4&and&Planet&6,&consecutively.&
4.& Draw&a&new&30^metre&line&extending&from&the&Sun&at&a&perpendicular&90^degree&
angle.&&Mark&circles&at&the&following&points:&9^metres,&15^metres,&21^metres,&26^
metres&and&30^metres.&These&points&will&represent&the&distant&stars,&label&them&
Star&A^E.&
ii.#Introduction#
1.& Ask&the&students&if&they&know&what&the&term&“parallax”&means.&Explain&that&it&is&
the&apparent&change&in&the&position&of&a&nearby&object&that&is&actually&caused&by&
the&movement&of&the&observer.&This&may&seem&like&a&foreign&concept&to&some&
students,&so&explain&that&this&is&something&that&you&use&constantly,&even&though&
you&might&not&notice&it.&&
PARALLAX: ACTIVITY GUIDE
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2.& Place&a&sticker&somewhere&in&the&room.&Ask&the&students&to&extend&their&arm,&
close&one&eye&and&cover&the&object&in&their&vision&with&their&thumb.&Then&
instruct&the&students&to&keep&their&hand&still&and&switch&eyes&(close&the&other&
eye).&&Ask&them&to&switch&back&and&forth&several&times.&Discuss&the&following&
questions:&
•
•
•
•
What&did&you&notice?&
Did&your&thumb&physically&move?&
How&did&your&thumb&appear&to&move?&
What&could&explain&your&thumb’s&apparent&movement?&
3.& Ask&the&students&to&move&their&thumb&close&to&their&face&while&switching&eyes.&
Ask&them&the&following&questions:&
•
•
•
How&does&the&apparent&motion&of&your&thumb&change?&
What&is&it&about&your&eyes&that&allows&you&to&see&this?&
If&you&could&make&your&arms&longer,&would&you&expect&the&apparent&shift&
to&increase&or&decrease?&Why?&
4.& Explain&to&your&students&that&the&change&in&their&thumb’s&apparent&position&
relative&to&the&background&object&is&due&to&a&change&in&the&viewer’s&position.&In&
this&case,&the&“viewer”&is&your&left&or&right&eye.&&The&few&centimetres&of&
separation&between&your&eyes&means&that&their&viewing&positions&are&different.&&
5.& Now&explain&to&your&students&that&parallax&allows&us&to&Bind&the&position&of&
distant&objects&such&as&stars,&since&just&like&our&eyes,&Earth&changes&position&
every&6&months&as&it&orbits&the&Sun.&&
Parallax&can&be&used&to&measure&the&distances&to&nearby&stars.&As&the&Earth&orbits&the&
sun,&a&nearby&star&will&appear&to&move&against&the&more&distant&background&stars,&in&
the&same&way&the&tree&appears&to&move&against&the&more&distant&mountains&in&the&
diagram&above.&&
Astronomers&can&measure&a&star's&position&twice&in&a&year,&with&a&6^month&gap&
between&the&Birst&and&second&measurement,&and&calculate&the&apparent&change&in&
position.&The&star's&apparent&motion&is&called&stellar&parallax.&
There&is&a&simple&relationship&between&a&star's&distance&and&its&parallax&angle:&
d+=+1/p+
The&distance&d&is&measured&in&parsecs,&the&parallax&angle&p&is&measured&in&arcseconds.&
The&radius&of&Earth’s&orbit&is&1&AU&(149,&598,&000&km).&
iii.#Outdoor#Activity#
PARALLAX: ACTIVITY GUIDE
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1.& Ask&students&to&gather&into&groups&(no&more&than&5&students&per&group)&and&
lead&them&outside&to&the&Parallax&Diagram.&&&
2.& Assign&each&team&to&one&of&the&planets&and&provide&them&with&an&Astrolabe,&
Instruction&Sheet&and&a&Worksheet.&
3.& Ask&the&groups&to&use&the&Parallax&Diagram&to&work&through&part&one&of&the&
Student&Worksheet&
4.& When&they&have&Binished&part&one&of&their&worksheets,&gather&the&groups&and&
head&back&into&the&classroom.&&
iv.#In#the#Classroom##
1.& Open&the&Parallax&table&spreadsheet&on&your&computer&and&use&a&projector&to&
allow&the&class&to&see&it&for&the&class&to&see.&&
2.& Ask&each&group,&one&at&a&time,&to&provide&the&parallax&angle&for&each&star&as&seen&
from&their&planet.&Leave&this&data&on&the&screen&for&the&class&to&see&and&ask&them&
to&complete&part&two&of&their&activity&worksheet.&
3.& Once&the&students&have&worked&through&the&second&part&of&their&worksheet&
move&on&to&the&discussion&section.&
v.#Discussion:#Results#
1.& Ask&the&students&what&they&notice&about&the&results&–&did&each&team&end&up&
the&same&distance?&Is&there&a&correlation&between&percent&error&and&
baseline&distance&(distance&from&the&Sun?)&
2.& If&each&group&measured&the&parallax&angle&with&the&same&amount&of&care,&
there&should&be&a&notable&trend&showing&teams&assigned&to&planets&further&
from&the&Sun&getting&more&accurate&results.&Ask&students&why&they&think&this&
is.&&&
The&answer&is&that&as&the&angle&gets&closer&to&90^degrees,&inaccuracies&are&
more&exaggerated&by&the&tangent&function.&This&means&that&a&very&long&
baseline&is&advantageous&for&parallax&measurements.&&
3.& Ask&students&what&is&the&longest&possible&baseline&for&Earth^based&parallax&
measurements?&&The&answer&is&2&AU,&or&twice&the&distance&between&the&
Earth&and&the&Sun.&
4.& Do&the&students&notice&a&correlation&between&the&distance&of&a&star&and&the&
percent&error?&&&
For&the&reason&stated&previously,&the&further&a&star&is&the&less&accurate&its&
distance&calculations&will&be&on&average.&&This&is&one&of&the&problems&with&
the&parallax&method&for&measuring&distances&in&the&cosmos.&&
5.& Ask&the&students&to&Bill&in&part&three&of&their&worksheet&before&discussing&the&
problems&with&parallax&in&further&detail.&
PARALLAX: ACTIVITY GUIDE
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Conclusion
Since&the&parallax&method&cannot&be&used&beyond&100&parsecs,&have&your&students&
look&up&other&possible&techniques&for&measuring&more&distant&astronomical&objects.&
Examples&include&the&use&of&Cepheid&Variable&stars&and&Type&1a&supernovae&for&
nearby&galaxies&and&redshift&for&those&further&away.&
The&Cosmic&Distance&Ladder&activities&part&2&and&3&will&go&into&further&detail&on&the&
use&of&Cepheid&Variables&and&type&1a&Supernovae&for&measuring&distances.&(LINK)&
Evaluation
(Optional)Discussion:#Problems#with#Parallax:#
It&sounds&fairly&easy&to&measure&the&distance&to&another☆&just&make&two&
measurements&of&its&position&six&months&apart.&&
In&practice,&however,&it&is&very&difBicult.&The&Birst&successful&measurement&of&stellar&
parallax&came&more&than&two&hundred&years&after&the&invention&of&the&telescope.&
Ask&your&students&to&discuss&some&possible&difBiculties&of&measuring&cosmic&
distances&using&parallax.&Ensure&they&mention&the&following&points:&
Parallax&shifts&are&always&small.&&
Parallax&shift&is&even&smaller&than&the&apparent&size&of&the&star.&In&additional,&
starlight&is&refracted&by&Earth's&atmosphere&and&causes&the&star&to&appear&blurred.&
Determining&the&position&of&a&star,&plus&that&of&several&reference&stars&in&the&same&
Bield,&to&a&very&small&fraction&of&this&blurry&dot&is&not&an&easy&task.&&
All&stars&in&a&Bield&exhibit&parallax.&&
In&practice,&astronomers&usually&measure&the&shift&of&one&star&in&an&image&relative&
to&other&stars&in&the&same&image.&However,&as&the&Earth&moves&from&one&side&of&the&
Sun&to&the&other,&we&will&see&all&the&stars&in&the&Bield&shift,&not&only&those&of&interest.&
Astronomers&must&pick&out&a&set&of&reference&stars&that&happen&to&be&much&farther&
away&than&the&target&star.&Distant&stars&will&shift&by&a&much&smaller&angle,&so&by&
measuring&the&position&of&the&nearby&target&star&relative&to&those&distant&ones,&it&is&
possible&to&detect&its&minute&shift.&
Limitations&of&Distance&Measurement&Using&Stellar&Parallax&
Parallax&angles&of&less&than&0.01&arcsec&are&very&difBicult&to&measure&accurately&from&
Earth&because&of&the&effects&of&the&atmosphere.&This&limits&Earth^based&telescopes&
to&measuring&the&distances&to&stars&about&1/0.01&or&100&parsecs&away.&Space^based&
telescopes&can&get&accuracy&to&0.001,&which&has&increased&the&number&of&stars&
whose&distance&could&be&measured&with&this&method.&However,&most&stars&even&in&
our&own&galaxy&are&much&further&away&than&1000&parsecs,&since&the&Milky&Way&is&
PARALLAX: ACTIVITY GUIDE
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about&30,000&parsecs&across.&The&next&section&describes&how&astronomers&measure&
distances&to&more&distant&objects.&
PARALLAX: ACTIVITY GUIDE
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The Cosmic Distance Ladder:
Parallax
Answer Sheet
Answers
Question 1: Planets further from the Sun yield more accurate results because the parallax
angle from these planets is further from 90-degrees.
Question 2: Stars closer to Earth yield more accurate results because the parallax angle for
these stars is further from 90-degrees.
Question 3.
❖ a. 1.300 pc
❖ b.#3.472#pc#
❖ c.#52.632#pc
❖ d.#3.35#pc
PARALLAX: ANSWER SHEET
1
The Cosmic Distance Ladder:
Standard Candles
Student Worksheet
STANDARD CANDLES: STUDENT WORKSHEET
1
Name:______________________
Date:_______________________
Instructions
1.###Begin#by#measuring#the#distance#between#your#planet#and#the#Sun.#Record#this#
on#the#corresponding#row#of#Table#1.#
2.# Now,#take#your#astrolabe#and#hold#it#with#the#Front#Mother#facing#towards#you.#
Standing#by#your#planet,#carefully#align#the#astrolabe#so#that#the#0degree#mark#
runs#along#the#baseline.#IMAGE#
3.# #Point#the#Rule#towards#your#star#as#precisely#as#possible.#Read#the#angle#from#
the#inner#ring#on#the#astrolabe’s#Mother#Front.#Record#the#angle#on#Table#1.#
This#is#the#parallax#angle,#p.##
Note:#Be#careful#to#read#the#angle#correctly.#Direct#the#sharp#point#of#the#Rule#
towards#the#star#and#read#the#angle#along#the#long#edge#of#the#Rule.#
4.# Repeat#this#process#and#note#down#the#parallax#angle#for#all#5#stars.#
Table#1:
Planet
Distance
from Sun
(Dp)
Star A (p)
Star B (p)
Star C (p)
Star D (p)
Planet 1
Planet 2
Planet 3
Planet 4
Planet 5
Planet 6
5.# Using#data#provided#by#your#classmates,#Qill#in#the#parallax#angle#for#stars#ARD#
from#each#planet.#
6.# You#will#now#calculate#the#distance#to#each#star#and#Qill#in#the#Table#2.#
Table 2:
STANDARD CANDLES: STUDENT WORKSHEET
2
Calculated distance to stars (d)
Planet
Distance from
Sun (Dp)
Star A (p)
Star B (p)
Star C (p)
Star D (p)
Planet 1
Planet 2
Planet 3
Planet 4
Planet 5
Planet 6
Average
Distance
7.# Calculate#the#percentage#error#for#each#star#and#Qill#in#Table#3#below.#
To#calculate#the#percentage#error#Qirst#Qind#the#different#between#the#actual#
distance#and#average#calculate#distance.#Divide#this#by#the#actual#distance,#and#
multiply#your#answer#by#100#to#convert#it#to#a#percentage.#
Table 3:
Star
Average Distance
(measured)
Actual Distance
Percentage Error
Star A
Star B
Star C
Star D
STANDARD CANDLES: STUDENT WORKSHEET
3
Questions
1.# Which#planets#yielded#the#most#accurate#results#on#average,#those#closer#to#
the#Sun#or#those#farther#away?#Why#do#you#think#this#was#the#case?#
2.# Which#stars#yield#the#most#accurate#results,#those#closer#to#us#or#those#
farther#away?#Why#do#you#think#this#is#the#case?#
In#actuality,#astronomers#use#the#angle#that#the#star#appears#to#move#compared#to#
more#distant#background#stars#as#Earth#orbits#the#Sun.#This#is#measured#size#
months#apart#to#get#the#greatest#angle,#then#divided#by#two#to#get#the#parallax#angle#
(p).#The#formula#they#use#to#express#the#simple#relationship#between#a#star's#
distance#and#its#parallax#angle#is:#
d!=!1/p!
The#distance#d#is#measured#in#parallax#seconds#or#parsecs,#the#parallax#angle#p#is#
measured#in#arcseconds#,#there#are#3600#arcseconds#in#1#degree.#The#radius#of#
Earth’s#orbit#is#1#AU#(149,#598,#000#km).#
3.# Using#the#parallax#angles#provided#below#calculate#the#distance#to#each#star#
in#parsecs.#Remember#that#astronomers#often#measure#parallax#in#units#
called#arcseconds.#(Show#your#work.)#
STANDARD CANDLES: STUDENT WORKSHEET
4
a.# Proxima#Centauri#has#a#parallax#of#0.769#
b.# Procyon#has#a#parallax#of#0.288#(3.472#pc)#
c.# Beta#Canis#Minoris#has#a#parallax#of#0.019#(52.632#pc)#
d.# Ross#128#has#a#parallax#of#0.298#(3.35#pc)#
STANDARD CANDLES: STUDENT WORKSHEET
5
The Cosmic Distance Ladder:
Standard Candles
Activity Guide
STANDARD CANDLES: ACTIVITY GUIDE
1
Introduction
Everyone(knows(that(a(candle(will(appear(dimmer(the(further(away(it(is,(and(this(is(
also(true(for(other(light(sources(—(such(as(stars.(In(this(activity(students(will(learn(how(
the(relationship(between(brightness(and(distance(can(be(used(to(determine(the(vast(
distances(in(the(Universe.((
Materials List
•
•
•
Standard(Candles:(Student(Worksheet((doc)(
Cepheid(Variables(data(Files(
Computer(with(spreadsheet(software(installed((per(student)(
Background Science
While(parallax(is(used(to(calculate(the(cosmic(distance(to(relatively(nearby(stars,(other(
methods(must(be(used(for(much(more(distant(objects,(since(their(parallax(angle(is(too(
small(to(measure(accurately.(One(comes(from(using(the(inherent(brightness(of(an(
object.(The(luminosity(of(some(astronomical(bodies(is(known(well(enough(to(be(able(to(
calculate(their(distance.(
A(star’s(apparent'magnitude(is(how(bright(a(star(appears(in(the(sky,(this(is(a(function(
of(both(its(actual(light(output,(known(as(it’s(absolute'magnitude,(and(it’s(distance(
from(the(observer.(If(we(have(values(for(the(former(two,(we(are(able(to(calculate(the(
latter.(
A(100(watt(light(bulb(right(next(to(you(will(appear(very(bright,(whereas(a(100(watt(
light(bulb(in(the(next(room(to(you(will(appear(much(fainter.(If(you(know(the(distance(to(
the(bulb(and(measure(its(apparent(brightness,(we(can(calculate(is(wattage.(Likewise,(if(
you(know(the(wattage(of(the(bulb(and(measure(the(observed(brightness,(you(can(
calculate(the(distance(to(the(bulb.(This(method(can(also(be(applied(to(stars,(if(you(
determine(the(apparent(magnitude(of(a(star(and(you(know(its(absolute(magnitude,(you(
can(calculate(the(distance.(
Astronomical(objects(for(which(the(absolute(magnitude(is(well(known(are(referred(to(
as(“standard(candles”.(One(commonly(used(type(of(standard(candle(are(Cepheid(
variables(–(a(type(of(star(for(which(the(luminosity(Fluctuates(over(time.((
There(is(a(direct(relationship(between(a(Cepheid(Variable’s(luminosity(and(the(period(
over(which(it(oscillates.(In(the(early(1900s(Henrietta(Levitt(and(Harlow(Shapley(noted(
that(as(the(period(of(Cepheid(variables(increased,(the(absolute(magnitude(also(
increased.(This(discovery(led(to(the(establishment(of(the(PeriodVLuminosity(law(that(is(
now(used(to(calculate(distances(to(Cepheid(variable.(The(PeriodVLuminosity(law(is:(
STANDARD CANDLES: ACTIVITY GUIDE
2
M'='32.81'log(P)'3'1.43'
Where(M(is(the(absolute(magnitude(and(P(is(the(period.(
Once(M(is(known,(the(distance(modulus(function(can(be(used(to(calculate(the(distance(
to(the(Cepheid(variable(in(parsecs((pc):(
Rearranged:(
m'3'M'='5'log'd'–'5'
d'='10(m$%$M'+'5)/5'parsecs'
Cepheid$Variable$Data$
The(data(used(in(this(activity(comes(from(the(American(Association(or(Variable(Star(
Observers((aavso.org).(The(data(for(each(of(the(variable(stars(chosen(for(this(activity(
was(collected(of(many(years,(a(time(that(is(expressed(on(the(Files(using(Julian'Day'
Number'(JDN).((
Julian(Day(Number(is(often(used(by(astronomers,(it(is(the(continuous(count(of(days(
since(the(beginning(of(the(Julian(Period(on(the(date(January(1,(4713(BC,(
Julian(calendar.(
For(example,(the(Julian(day(number(for(the(day(starting(at(12:00(UT(on(January(1,(
2000,(was(2,451,545.
Astronomical Distances
Word
Definition
Astronomical Unit (AU)
An(astronomical(unit(is(the(radius(
of(Earth’s(orbit(around(the(Sun.(1(
AU(=(149(597(871(km.
Light Year (ly)
A(light(year(is(a(unit(of(
measurement(used(for(the(vast(
distances(in(the(cosmos.(One(light(
year(is(equal(to(the(distance(that(
light(can(travel(through(the(
vacuum(of(space(in(one(year.(1(ly(=(
63241(AU
Parsec (pc)
A(parsec(another(unit(of(
measurement(for(distances(in(the(
Universe.(Parsec(is(a(combination(
of(two(words,(parallax((par)(and(
arc(second((sec).(!1pc(=!3.26(light(
years.(
STANDARD CANDLES: ACTIVITY GUIDE
3
Instructions
In(this(activity(students(will(learn(how(the(periodical(change(in(luminosity(of(variable(
stars(can(be(used(to(calculate(their(distance(and(calculate(the(distance(to(six(Cepheid(
variable(stars(using(data(from(the(American(Association(of(Variable(Star(Observers(
(AAVSO).(
i.!Introduction:!
Begin(the(activity(by(discussing(ways(to(determine(distances(in(the(cosmos.(Explore(
the(students’(ideas(on(how(they(might(calculate(distances(to(an(object(without(the(
ability(to(travel(to(it.((
Introduce(the(students(to(the(concept(of(Standard(Candles,(speciFically(focussing(on(
Cepheid(Variables((see(Background'Information).(Be(sure(to(explain(the(following(
terms:(absolute'magnitude,'apparent'magnitude,'period3luminosity'function'and'
distance'modulus.!
ii.!Activity:!
1.( Each(student(will(need(a(computer(with(spreadsheet(software(installed(
(preferably(Excel).((
2.( Provide(the(selection(of(Cepheid(Variable((CV)(data(sets(to(students(and(ask(
them(to(choose(one,(ensuring(that(each(CV(on(the(list(is(chosen(by(at(least(one(
student.((
3.( Ask(the(class(to(work(individually(through(questions(1V14(on(the(student(
worksheet(provided.(
4.( Once(the(students(have(completed(questions(1V14(on(the(student(worksheet,(
open(the(Teacher’s(Table(document(in(your(spreadsheet(software(and(5(CV.(
5.( Leave(the(completed(table(project(for(the(class(to(see.(Ask(them(to(refer(back(to(
their(worksheets(and(work(through(questions(15V20.(
Evaluation
The(students(now(have(an(understanding(of(one(method(astronomers(can(use(to(
measure(distances(in(the(cosmos.(However,(Cepheid(variable(stars(are(only(useful(if(
their(luminosity(and(period(can(be(measured,(i.e.(if(they(are(visible.(Past(30(Mpc,(this(
method(is(no(longer(viable.((
•
Ask(your(students(if(they(can(think(of(any(astronomical(objects(that(could(be(
used(to(in(a(similar(way(to(Cepheid(variables.(
Some(other(astronomical(objects(that(can(be(used(in(a(similar(way(to(Cepheid(varaibles(
are(Type(1a(supernovae.(All(Type(1a(supernovae(are(formed(from(the(same(physical(
STANDARD CANDLES: ACTIVITY GUIDE
4
process((a(white(dwarf(becoming(too(massive(causing(a(chain(reaction(leading(to(the(
explosion)(and(thus,(have(about(the(same(absolute(magnitude.(This(means(that,(like(
with(Cepheid(Variables,(they(work(as(Standard(Candles(and(once(their(apparent(
magnitude(is(known,(their(distance(can(be(calculated((see(Hubble(Diagram(activity).(
Cepheid Variable Fact Sheet
This table provides answers to the activity, it is not to be provided to students but can be
used to gauge the accuracy of their results. Please note that the student answers are not
expected to be exact.
Name
Constellation
AH Vel
Vela
l Car
Max Mag (V)
Min Mag (V)
Period (days)
Distance (pc)
5.5
5.89
4.221
446
Carina
3.28
4.18
35.535
480
RS Pup
Puppis
6.52
7.67
41.443
1910
X Cyg
Cygnus
5.85
6.91
16.386
700
TT Aql
Aquila
6.46
7.7
13.754
6562
FF Aql
Aquila
5.18
5.68
4.470
413
STANDARD CANDLES: ACTIVITY GUIDE
5
The Cosmic Distance Ladder:
Standard Candles
Student Worksheet
STANDARD CANDLES: STUDENT WORKSHEET
1
Name:______________________
Date:_______________________
Instructions
1.# You#will#have#6#.txt#1iles.#Each#one#contains#data#for#a#single#Cepheid#variable#
(CV)#collected#by#the#AAVSO#(aavso.org).#Each#1ile#contains#information#on#the#
change#in#magnitude#(brightness)#of#a#variable#star#over#time.#The#time#will#be#
shown#using#Julian#Date,#see#Background#Science#for#more#information).##
2.# Choose#one#Cepheid#Variable#from#the#6#options.#
3.# Open#the#spreadsheet#software#of#your#choice#(preferably#Excel#as#this#
instruction#sheet#is#speci1ically#designed#for#excel).##
4.# Go#to#File>Open#and#select#your#chosen#Cepheid#Variable#1ile.#
5.# On#the#1irst#window#that#appears#select#‘Delimited’,#as#shown#on#the#image#
below.#We#will#be#using#commas#to#separate#our#columns.#Click#Next.
#
6.#On#the#next#window#that#appears,#uncheck#the#box#labelled#Tab#and#select#
Comma#as#indicated#on#the#screenshot#below.##This#will#put#all#values#separated#
by#a#comma#into#a#separate#column.#Click#Next.
STANDARD CANDLES: STUDENT WORKSHEET
2
#
#
7.#Your#1ile#will#now#open#with#the#data#separated#into#distinct#columns.#We#will#
only#be#using#the#1irst#two#columns#for#this#activity:#Julian#Date#(JD)#and#
Magnitude#(Mag).##To#avoid#confusion#you#can#now#delete#all#other#columns.##
STANDARD CANDLES: STUDENT WORKSHEET
3
8.# The#Julian#Date#provides#the#day#to#the#left#of#the#decimal#point#and#the#time#to#
the#right.#We#only#want#to#plot#days,#so#delete#all#duplicated#days.#Do#this#by#
1irst#highlighting#your#JD#column,#then#rightWclick#on#the#column#and#selecting#
Format3Cells.#Next#select#Number3and#lower#the#number#of#decimal#places#to#
0.
9. To#delete#all#duplicates,#highlight#the#JD#column#and#go#to#Data3>3Remove3
Duplicates.3Ensure#you#group#the#two#columns#before#doing#this#so#the#
corresponding#magnitude#is#also#deleted.#Do#this#by#selecting#the#two#columns#
and#clicking#Data>Group>Columns
10.#You#will#notice#there#are#tens#of#thousands#of#days#worth#of#observational#data#
in#each#1ile.#We#will#only#need#around#200#days.#Choose#a#selection#of#100#
consecutive#days#from#your#data.#(Data#collection#started#early#in#the#1900s#so#
the#instruments#may#not#have#been#as#accurate#as#more#recent#data#further#
down#in#the#spreadsheet).#Delete#all#additional#rows,#ensuring#you#have#enough#
data#to#make#a#decent#plot.#
11.#Now#highlight#both#columns#and#select#Chart3>3Scatter3>3Marked3Scatter3to#
create#a#scatter#graph#showing#your#star’s#magnitude#over#time.#
12.#You#may#need#to#edit#the#axes#range#of#your#plot#to#better#suit#your#data.#To#do#
this#select#Chart3>3Chart3Layout3>3Axes3>3Axis3Options3>3Scale.#
STANDARD CANDLES: STUDENT WORKSHEET
4
13.##Change#the#minimum#and#maximum#to#suit#your#data#after#looking#at#your#
minimum#and#maximum#data#values. Your#light#curve#should#look#similar#to#the#
example#below:#
FF*Aql*Light*Curve*
6$
Magnitude*(m)*
5.8$
5.6$
5.4$
5.2$
5$
4.8$
2451340$
2451360$
2451380$
2451400$
2451420$
2451440$
2451460$
JD*
14.#Now,#use#your#data#to#1ind#the#following#values#and#1ill#in#the#table#at#the#end#of#
this#worksheet:#
o What#is#the#average#period#of#your#Cepheid#Variable?#
o What#is#the#maximum#magnitude?#
o What#is#the#minimum#magnitude?#
15.#Work#out#the#mean#apparent#magnitude#(m)#for#your#Cepheid#variable#to#3#
decimal#places#and#record#the#result#on#your#table.#
16.#Using#a#calculator#1ind#log(P)#for#your#Cepheid#variable,#where#P#is#its##period.#
17.#Your#teacher#will#now#collect#the#values#for#your#Cepheid#variable#and#share#
values#for#the#other#members#of#the#group.#Add#their#results#to#your#table.##
18.#Copy#your#data#from#the#table#below#into#your#spreadsheet#to#create#a#graph#
showing#magnitude#vs.#log#period#for#all#6#Cepheid#variables#
19.#Add#a#trend#line#through#your#data.#Do#this#by#selecting#Chart3>3Chart3Layout3>3
Trendline.#Check#that#none#of#your#data#points#vary#too#much#from#the#
trendline,#if#there#are#any#that#are#too#far#from#the#trendline#ensure#you#check#
the#values#again.##This#trendline#represents#the#periodWluminosity#relationship#
for#the#Cepheid#variables#you#are#working#with.#
19.#Using#the#period#(P)#you#can#calculate#the#absolute#magnitude#(M)#for#each#
Cepheid#Variable#using#the#function#below,#known#as#the#PeriodWLuminosity#
function:#
M3=3E2.813log(P)3E31.433
The#periodWluminosity#law#can#be#used#because#there#is#a#direct#relationship#
between#a#Cepheid#Variable’s#luminosity#and#the#period#over#which#it#oscillates#
–#when#the#period#increases#so#does#the#absolute#magnitude.#You#should#be#able#
to#see#this#on#your#second#graph.#
STANDARD CANDLES: STUDENT WORKSHEET
5
Using#the#luminosity#function,#calculate#M"for#your#6#stars#and#enter#the#values#
on#to#your#spreadsheet.#
20.#Once#M#is#known,#the#distance#modulus#function#can#be#used#to#calculate#the#
distance#to#the#Cepheid#variable#in#parsecs#(pc):#
Rearranged:#
m3E3M3=353log3d3–353
d3=310(m$%$M3+35)/53parsecs3
Where"m#is#apparent#magnitude,#M#is#absolute#magnitude#and#d#is#distance#in#
parsecs.#
21.#Using#the#distance#modulus#equation#calculate#the#distance#to#each#Cepheid#
variable#in#parsecs#and#record#your#results#in#the#table#on#your#spreadsheet.#
Congratulations,#you#have#now#successfully#calculated#the#distance#to#six#Cepheid#
Variables!##
Results Table
CV Name Period
(days)
Min Mag. Max
Mag.
STANDARD CANDLES: STUDENT WORKSHEET
Mean
Mag (m)
Log(P)
Absolute
Mag. (M)
Distance
(d)
6
LCOGT Activity Form
Title: Measuring the Age of the Universe
Keywords:
Hubble diagram, distance, redshift, radial velocity, expansion, inflation, Universe, Big Bang,
spectroscopy, Type Ia supernovae, Doppler Effect, Hubble Constant, Age of the Universe
Age: Level: Secondary School
Time: 1.5 hours
Group: Individual
Supervised: Supervised
Cost: Low (< ~5 EUR)
Location: Indoors
Language: EN
Brief description:
The discovery of the expanding Universe was one of the greatest revelations in astronomy.
During this activity students will relive Hubble’s monumental discovery by using real supernova
spectra to create a famous Hubble Diagram.
List of Materials:
Per student:
• One computer with Internet connection
• Hubble Diagram Worksheet (.xls)
• Hubble Diagram Instruction Sheet
Goals: You will use real spectra to create you own Hubble Diagram by calculating the distance
and redshift to a selection of Type Ia supernovae. You will use the Hubble Diagram to explore
the expansion of the Universe, work out the Hubble Constant and calculate the current age of
the Universe.
Learning Objectives:
•
•
•
•
•
•
Understand and demonstrate the concept of the expanding Universe using the famous
Hubble Diagram,
Become familiar with the study of spectroscopy in astronomy and how it can be used to
explore the Universe,
Conduct scientific analysis and calculations, and create plots using real astronomical
data.
Understand and practise the use of Type Ia supernovae as standard candles to measure
distances in the Universe,
Use real supernovae spectra to calculate redshift and radial velocity,
Analyse the Hubble Diagram to work out the Hubble Constant and calculate the current
age of the Universe.
Background Information:
i. Standard Candles, Type Ia supernovae and the Distance Modulus
Measuring distances in space is a daunting task. One method is to use an object with a known
absolute magnitude (M); we call these Standard Candles. Type Ia supernovae are standard
candles.
There are two classes of supernovae, Type I and Type II. For this activity we will be using Type
Ia supernovae only. Type Ia supernovae are very important in astronomy as they offer the most
reliable sources for measuring cosmic distances up to and beyond 1000 mega parsecs (Mpc). A
parsec (pc) is a unit of length used to measure the distances to objects in space. One parsec is
equal to 3.08567758 × 1016 metres.
Type Ia supernova are thought to form when a white dwarf star accretes enough mass from a
companion star to exceed the upper limit of 1.44 solar masses, this causes the star to explode
as a supernova.
This model implies that all Type Ia supernovae start with essentially the same mass and
therefore the energy output from the resulting supernova should always be the same. Of course,
it is not quite that simple, but all Type Ia supernova do seem to have similar light curves and can
therefore be related to the same common template. There is an average maximum absolute
magnitude that these supernovae can reach in the visible wavelength band B. The peak
brightness is: -19.6 mag.
If we also know the apparent magnitude for a Type Ia supernova, we can just the distance
modulus to calculate its distance from earth. The distance modulus is the difference between
the apparent magnitude (m) and the absolute magnitude (M) of an astronomical object. It can be
used to provide the distance (d) in parsecs. Once m is known, the distance modulus function
can be used to calculate the distance to the supernova in parsecs (pc):
d = 10(m - M + 5)/5 pc
ii. Spectra, Redshift and Radial Velocity
To create a Hubble Diagram, the redshift and velocity of your object is needed, as well as the
distance. To find the redshift of an object astronomers use spectroscopy. Spectroscopy is a
branch of science that is concerned with the investigation and measurement of the spectrum of
light produced when matter interacts with electromagnetic radiation. In astronomy, spectroscopy
is a very useful tool because telescopes can easily measure the spectra of far away objects,
such as stars and galaxies.
We measure the spectra of luminous objects by attaching a special instrument to a telescope,
known as a spectrograph. Within the spectrograph, light is passed through a series of prisms
and mirrors in order to split the light into its different component wavelengths (its spectrum). The
intensity of those different wavelength components is then measured to create a spectrum. The
spectrum can then be investigated in detail to discover useful information about the target such
as chemical composition, Doppler shift and velocity.
The Doppler Effect, such as redshift, occurs due to an apparent shift in the wavelength and
frequency of a wave due to relative motion between the source and the observer. For example,
we experience the Doppler Effect when an ambulance drives past with its sirens on:
- As the ambulance (the source) approaches you (the observer), the sound wave is
compressed (bunched up) and the resulting sound is higher pitched (shorter wavelength (λ))
- Once the ambulance has passed, the wave is stretched and the pitch sounds lower
(longer λ).
Illustrating demonstrating the Doppler effect, with the shortest wavelength on the left and longer
wavelengths on the right.
The same effect is observed with light, when an object is moving towards or away from
observers located on the Earth. We can detect this motion in the spectrum of the star.
If the star is moving away from the Earth, the light waves are stretched out and the whole
spectra experiences a shift towards the red end of the spectrum – a red shift. If the star is
moving towards the Earth, the light waves are bunched up and the spectra experiences a shift
towards the blue end of the spectrum – a blue shift.
We can measure comparing an observed wavelength to a rest wavelength. For example, the
rest wavelength for Hα (when there is no Doppler Effect) is 6563Å, if it is redshifted the
observed wavelength will be longer. If both the observed and rest wavelengths are known, these
can be used to calculate the object’s redshift can be calculated, where z = redshift:
z = (observed wavelength/rest wavelength) – 1
v. The Hubble Diagram and Hubble’s Law
The idea that we live in an expanding Universe was one of the most unexpected and important
discoveries of the 20th century. Until then, everyone, including astronomers, had assumed that
the Universe was a stable, unchanging stage on which astronomical events played themselves
out. But then, in the early 1900s scientists made several discoveries that defied easy
explanation. These discoveries came together in the mind of an astronomer named Edwin
Hubble, who explained all of them in 1929 with the expanding universe theory.
The discovery of the expanding universe was, like more discoveries, a combined effort. In the
early 1900s an American astronomer called Vesto Slipher studied the spectra of a number of
“fuzzy objects” believed to be nebula (we now know many of them were separate galaxies). He
noticed that many of them were redshifted, which meant they were moving away from us. Some
of them were moving away from us at amazing speeds – for example the object we now know of
as the Sombrero Galaxy was moving away from us at speeds of 4 million km per hour.
While Slipher made this discovery, in another part of the world Edwin Hubble was using the
world biggest telescope (at the time) to measure the distance to astronomical object using
creative new methods. Hubble then compared his measurements of distance, to Slipher’s
measurements of redshirt and plotted the results – this plot is now known as the Hubble
Diagram.
The Hubble diagram demonstrated that a galaxy’s redshift increased linearly with its distance
from Earth. The farther away a galaxy is, the faster it moves away from us.
The simplest explanation for Hubble’s observation was that the entire universe is expanding,
just as Einstein's equations predicted it should. If the universe is expanding, then at some time
in the past, it must have started from a single point - an idea known as the Big Bang. Hubble's
discovery, and the later development of the big bang theory, changed astronomy forever.
In this activity students will follow the steps on the student handout to plot the famous Hubble
Diagram to demonstrate the expanding Universe, beginning by determining the redshift for 10
Type Ia supernovae, from which they will derive the radial velocity for each supernova. They will
then calculate the distance to each supernova and use the two values to plot the famous Hubble
Diagram.
Once they have plotted the Hubble Diagram, your students will study their results and hopefully
come to the conclusions that Hubble reached, now known as Hubble’s Law. Hubble's law is the
name for the observation in physical cosmology that: (1) objects observed in deep space
(extragalactic space which is more than 10 Mpc) are found to have a Doppler shift that can be
used to calculate their relative velocity away from the Earth; and (2) that the velocity, of various
objects receding from the Earth, is approximately proportional to their distance from the Earth.
This is normally interpreted as a direct observation of the expansion of the Universe.
The expansion of the Universe was first proposed by Georges Lemaître in 1927, he also
suggested an estimated value of the rate of expansion, now called the Hubble Constant. Hubble
also calculated a value for the Hubble Constant just two years later, he put it at around 500
km/s/Mpc. We now know this was incorrect. In this activity your students will also calculate the
Hubble Constant based on their own Hubble Diagram. From the Hubble Constant,
Full Description:
In this activity we will begin by plotting a famous Hubble Diagram to demonstrate the expanding
Universe. The first step is to find spectra for ten different Type Ia supernovae and work out the
redshift for each. Using the redshift, you will derive the radial velocity for each supernova. Next,
you will calculate the distance to each of your objects. Using these two values you will plot the
famous Hubble Diagram.
Calculating Redshift
An apparent shift in the wavelength and frequency of a wave due to relative motion between the
source and the observer is called the Doppler Shift. This effect is observed with light, when an
object (in this case a supernova) is moving towards or away from observers located on the
Earth. We can detect this motion in the spectrum of the star.
If the star is moving away from the Earth, the light waves are stretched out and the whole
spectra experiences a shift towards the red end of the spectrum – a red shift. If the star is
moving towards the Earth, the light waves are bunched up and the spectra experiences a shift
towards the blue end of the spectrum – a blue shift.
You will now look at the spectra of a number of Type Ia supernova and calculate the Doppler
shift using the rest wavelength (the observed wavelength of a spectral line when there is no
relative motion between the observer and the source of the radiation) vs. to the observed
wavelength of spectral features.
1. Begin by opening the Hubble Diagram worksheet in the spreadsheet software of your
choice. Then open your web browser and go to: http://wiserep.weizmann.ac.il/spectra/list
This website is an archive containing spectra for a number of different types of astronomical
object collected by a range of instruments. For this activity we will be using spectra for Type Ia
supernovae collected by the LCOGT FLOYDS spectrograph and the STIS spectrograph aboard
the Hubble Space Telescope.
2. You can ignore all fields on the first page apart from ‘Obj Type’. Select ‘SN Ia’ from the
dropdown list.
3. Next, select FTS - FLOYDS-S from the Instrument list; this is an instrument upon one of the
LCOGT telescopes in Austrailia. Then press the
button.
4. You will see a supernova spectrum. Careful analysis of spectra like this can reveal many
things about as star, including temperature, density and chemical composition. We will be
using measurements of wavelength to calculate how the supernova is moving relative to us.
5. Below the graph you will notice two values for the wavelength (WL): observed and rest.
These values show us how the supernova is moving relative to us. The wavelength is shown
in a unit called Ångström (Å).
6. Hover your mouse over a defined peak on the graph and make a note of the observed
wavelength and rest wavelength values on your worksheet.
7. To the right of your graph you will notice check boxes for a number of elements: hydrogen,
helium, oxygen and sodium. When you click a box it will show you the emission or
absorption features for each element.
Take some time to explore these elements and find out which are present in each galaxy,
the more defined the feature, the heavier the presence of these elements. Which element is
most abundant in this supernova? Note down your answer in the ‘Element’ column on your
worksheet.
8. You will need to return to step 1, but this time you will select HST – STIS from the
Instrument dropdown list. Then choose ten supernovae from the list. Note down the name
of each in the table provided.
Do not select the following supernovae:
• PTF10pdf
• PTF10zdk
• PTF10acdh
• Sn2010kg
• Sn2010fe/PTF11kly
Note: Be careful to ensure that you have ten different supernova spectra and have not
duplicated any, as there are multiple spectra for some supernovae.
9. Note down the rest wavelength and observed wavelength of a chosen peak for each
supernova.
10. (Optional) You can try to identify the element responsible for the peak you use for your
wavelength values. If you can successfully identify the element, make a note of it on your
table.
Calculating Redshift and Radial Velocity
1. You can now calculate how much each supernova is shifted and the speed it is moving
away from or toward us (its Radial Velocity) in the following way. Where z is redshift:
z = (observed wavelength/rest wavelength) – 1
2. Round your answers to 3 decimal places and note them on your worksheet where indicated.
3. Next you will derive the velocity for each supernova using the redshift values you have
collected and the simple equation:
radial velocity = redshift × speed of light
Given that the speed of light is approximately 300,000 km/s, you can calculate the radial
velocity for each of your supernovae.
For example, if a supernova has a redshift of 0.025, then the radial velocity is given by
radial velocity = 0.025 × 300000 = 7500 km/s
4. Once the redshift has been calculated for each of your supernovae, add it to your
spreadsheet.
Note: This equation only works for redshift smaller than z = 1.0.
A. Discussion question: You may have noticed that all your redshift figures are positive. What
would it mean if your redshift were negative?
Calculating Distances
Measuring distances in space is a daunting task. One method of measuring distances is to use
an object with a known absolute magnitude (M), we call these Standard Candles. Type Ia
supernovae are standard candles.
A star’s apparent magnitude (m) is how bright a star appears in the sky, this is a function of both
its actual light output, known as it’s absolute magnitude (M), and it’s distance from the observer.
If we have values for the former two, we are able to calculate the latter.
Type Ia supernovae happens when a white dwarf star reaches 1.4 solar masses, this causes a
nuclear chain reaction to occur that causes the star to explode. This chain reaction always
happens in the same way and at the same mass, which means the brightness of Type Ia
supernovae is always the same, a peak light output approximately equivalent to an absolute
blue sensitive magnitude of -19.6.
Once m is known, the distance modulus function can be used to calculate the distance to the
Type Ia supernova in parsecs (pc):
m - M = 5 log d – 5
Rearranged:
d = 10(m - M + 5)/5 pc
1. To calculate the distance to your Type Ia supernovae, you need to note down the
maximum B-band magnitude. Open the ‘B-band Max Mag’ sheet on your spreadsheet.
The data in this table is from a scientific paper called “Hubble Space Telescope studies
of low-redshift Type Ia supernovae: Evolution with redshift and ultraviolet spectral trends“
which you can find at http://arxiv.org/pdf/1205.7040.pdf
2. There is a column in the table called B-band max (mag) this shows the maximum
apparent magnitude (m) for each supernovae. Make a note of this value down on your
table.
Note: if the table does not include the B-band maximum magnitude for your chosen
supernova, return to Calculating Redshift step 7 and select only supernovae which are
on this table.
3. Now that you have the apparent magnitude you can use the distance modulus to
calculate the distance to each supernova in parsecs.
Plotting a Hubble Diagram
1. Once you have worked through all of the previous steps you will have all you need to
plot a Hubble diagram. A Hubble diagram is a plot of radial velocity against distance. The
graph can be created using the chosen spreadsheet software.
2. The radial velocities should be plotted in kilometres per second on the Y-axis against
distance in mega parsecs (Mpc) on the X-axis. If you are not familiar with plotting graphs
in your spreadsheet software, you can use the Blank Graph template to plot the data
points by hand.
3. Once you have plotted the data for your supernovae, draw a straight line through them,
going through as many points as possible as shown in the example below. The line
should also go through zero.
Estimating the Hubble Constant
This line allows you to make an estimate of the Hubble constant. The Hubble constant is the
gradient of this line.
The Hubble constant (H0) is one of the most important numbers in cosmology because it can be
used to estimate the size and age of the Universe. The Hubble constant indicates the rate at
which the Universe is currently expanding, so although it is called a ‘constant’ it changes with
time.
The Hubble constant is give by:
H0 =v/d
Where v is the radial velocity, d is the distance and the 0 denotes that the number derived is the
Hubble constant at the present time. The value for the Hubble constant is given in kilometres
per second per mega parsec (Mpc).
1. Now, calculate the Hubble Constant. Simply pick any point on the line and read off the
position along the y-axis and the x-axis for that point and then divide them as shown
above to find out the Hubble constant in terms of kilometres per second per mega
parsec. Add the value to your worksheet where indicated.
You’ve now successfully determined the rate of the expansion of the Universe!
Conclusions:
Now that you have a value for the Hubble constant, one simple thing you can do to wrap up the
activity is estimate the age of the Universe. This can be done by finding the inverse of the value
of Ho
If the Universe has been expanding at a constant speed since its beginning, the Universe's age
would simply be 1/Ho
1. Calculate the age of the Universe using your value for the Hubble Constant.
2. Next, multiply you answer by 3.09 x 1019 km/Mpc to cancel the distance units. You now
have the age of the Universe in seconds!
3. Divide this number by the number of seconds in a year: 3.16 x 107 sec/yr
4. Now you have worked out the age of the Universe using real observational data!
Evaluation:
Ask the students to answer the following questions.
1. Compare your maximum age for the Universe to:
a. the age of the Sun (~5 billion years)
b. the age of the oldest stars in the Milky Way (~12.5 billion years)
2. What would happen to the calculated age of the Universe if the Hubble constant were:
a. larger than you found
b. smaller than you found
3. Why does the best-fit line to your data need to go through the origin of your graph?
Where is this "origin" located in the Universe?
4. Theoretically, your plot should be a straight line, but it probably isn't. Consider and write
down a few of the possible sources of error.
5. How does your result for the age of the Universe compare with other ages that have
been proposed? (The Hubble Constant is currently considered to be 67.8 km/s/Mpc.)
Advanced Activity Option:
Each of the following instruments offer 10 or more Type Ia supernovae spectra. For advanced
students you can assign each student on of the instruments (ensuring all instruments are
assigned) and ask them to calculate the redshift and radial velocity for ten supernovae from
theur instrument. The students will then need to research using the Internet to find the max Bband apparent magnitude, or the distance to each supernova themselves.
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Mayall - RC Spec
Magellan-Clay - MIKE
Magellan-Clay - LDSS-3
Magellan-Clay - LDSS-2
Lick 3m - UV Schmidt
HST – STIS
APO 3.5m – DIS
Additional information:
Connection to school curriculum: Country dependent
Additional files:
Hubble Diagram Worksheet.xls
License: CC
Measuring the Age of the Universe
Student Worksheet
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Name:______________________
Date:_______________________
In this activity we will begin by plotting a famous Hubble Diagram to demonstrate the
expanding Universe. The first step is to find spectra for ten different Type Ia supernovae
and work out the redshift for each. Using the redshift, you will derive the radial velocity for
each supernova. Next, you will calculate the distance to each of your objects. Using these
two values you will plot the famous Hubble Diagram.
Instructions
Calculating Redshift
An apparent shift in the wavelength and frequency of a wave due to relative motion
between the source and the observer is called the Doppler Shift. This effect is observed
with light, when an object (in this case a supernova) is moving towards or away from
observers located on the Earth. We can detect this motion in the spectrum of the star.
If the star is moving away from the Earth, the light waves are stretched out and the whole
spectra experiences a shift towards the red end of the spectrum – a red shift. If the star is
moving towards the Earth, the light waves are bunched up and the spectra experiences a
shift towards the blue end of the spectrum – a blue shift.
You will now look at the spectra of a number of Type Ia supernova and calculate the
Doppler shift using the rest wavelength (the observed wavelength of a spectral line when
there is no relative motion between the observer and the source of the radiation) vs. to the
observed wavelength of spectral features.
1. Begin by opening the Hubble Diagram worksheet in the spreadsheet software of your
choice. Then open your web browser and go to: http://wiserep.weizmann.ac.il/spectra/
list
This website is an archive containing spectra for a number of different types of
astronomical object collected by a range of instruments. For this activity we will be using
spectra for Type Ia supernovae collected by the LCOGT FLOYDS spectrograph and the
STIS spectrograph aboard the Hubble Space Telescope.
2. You can ignore all fields on the first page apart from ‘Obj Type’. Select ‘SN Ia’ from the
dropdown list.
3. Next, select FTS - FLOYDS-S from the Instrument list; this is an instrument upon one
of the LCOGT telescopes in Australia. Then press the
button.
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4. You will see a supernova spectrum. Careful analysis of spectra like this can reveal
many things about as star, including temperature, density and chemical composition.
We will be using measurements of wavelength to calculate how the supernova is
moving relative to us.
5. Below the graph you will notice two values for the wavelength (WL): observed and rest.
These values show us how the supernova is moving relative to us. The wavelength is
shown in a unit called Ångström (Å).
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6. Hover your mouse over a defined peak on the graph and make a note of the observed
wavelength and rest wavelength values on your worksheet.
7. To the right of your graph you will notice check boxes for a number of elements:
hydrogen, helium, oxygen and sodium. When you click a box it will show you the
emission or absorption features for each element.
Take some time to explore these elements and find out which are present in each
galaxy, the more defined the feature, the heavier the presence of these elements.
Which element is most abundant in this supernova? Note down your answer in the
‘Element’ column on your worksheet.
8. You will need to return to step 1, but this time you will select HST – STIS from the
Instrument dropdown list. Then choose ten supernovae from the list. Note down the
name of each in the table provided.
Do not select the following supernovae:
• PTF10pdf
• PTF10zdk
• PTF10acdh
• Sn2010kg
• Sn2010fe/PTF11kly
Note: Be careful to ensure that you have ten different supernova spectra and have not
duplicated any, as there are multiple spectra for some supernovae.
9. Note down the rest wavelength and observed wavelength of a chosen peak for each
supernova.
10. (Optional) You can try to identify the element responsible for the peak you use for your
wavelength values. If you can successfully identify the element, make a note of it on
your table.
Calculating Redshift and Radial Velocity
1. You can now calculate how much each supernova is shifted and the speed it is moving
away from or toward us (its Radial Velocity) in the following way. Where z is redshift:
z = (observed wavelength/rest wavelength) – 1
2. Round your answers to 3 decimal places and note them on your worksheet where
indicated.
3. Next you will derive the velocity for each supernova using the redshift values you have
collected and the simple equation:
radial velocity = redshift × speed of light
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Given that the speed of light is approximately 300,000 km/s, you can calculate the
radial velocity for each of your supernovae.
For example, if a supernova has a redshift of 0.025, then the radial velocity is given by
radial velocity = 0.025 × 300000 = 7500 km/s
4. Once the redshift has been calculated for each of your supernovae, add it to your
spreadsheet.
Note: This equation only works for redshift smaller than z = 1.0.
A. Discussion question: You may have noticed that all your redshift figures are positive.
What would it mean if your redshift were negative?
Calculating Distances
Measuring distances in space is a daunting task. One method of measuring distances is to
use an object with a known absolute magnitude (M), we call these Standard Candles. Type
Ia supernovae are standard candles.
A star’s apparent magnitude (m) is how bright a star appears in the sky, this is a function of
both its actual light output, known as it’s absolute magnitude (M), and it’s distance from the
observer. If we have values for the former two, we are able to calculate the latter.
Type Ia supernovae happens when a white dwarf star reaches 1.4 solar masses, this
causes a nuclear chain reaction to occur that causes the star to explode. This chain
reaction always happens in the same way and at the same mass, which means the
brightness of Type Ia supernovae is always the same, a peak light output approximately
equivalent to an absolute blue sensitive magnitude of -19.6.
Once m is known, the distance modulus function can be used to calculate the distance to
the Type Ia supernova in parsecs (pc):
m - M = 5 log d – 5
Rearranged:
d = 10(m - M + 5)/5 pc
1. To calculate the distance to your Type Ia supernovae, you need to note down the
maximum B-band magnitude. Open the ‘B-band Max Mag’ sheet on your
spreadsheet. The data in this table is from a scientific paper called “Hubble Space
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Telescope studies of low-redshift Type Ia supernovae: Evolution with redshift and
ultraviolet spectral trends“ which you can find at http://arxiv.org/pdf/1205.7040.pdf
2. There is a column in the table called B-band max (mag) this shows the maximum
apparent magnitude (m) for each supernovae. Make a note of this value down on
your table.
Note: if the table does not include the B-band maximum magnitude for your chosen
supernova, return to Calculating Redshift step 7 and select only supernovae which
are on this table.
3. Now that you have the apparent magnitude you can use the distance modulus to
calculate the distance to each supernova in parsecs.
Plotting a Hubble Diagram
1. Once you have worked through all of the previous steps you will have all you need
to plot a Hubble diagram. A Hubble diagram is a plot of radial velocity against
distance. The graph can be created using the chosen spreadsheet software.
2. The radial velocities should be plotted in kilometres per second on the Y-axis
against distance in mega parsecs (Mpc) on the X-axis. If you are not familiar with
plotting graphs in your spreadsheet software, you can use the Blank Graph
template to plot the data points by hand.
3. Once you have plotted the data for your supernovae, draw a straight line through
them, going through as many points as possible as shown in the example below.
The line should also go through zero.
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Estimating the Hubble Constant
This line allows you to make an estimate of the Hubble constant. The Hubble constant is
the gradient of this line.
The Hubble constant (H0) is one of the most important numbers in cosmology because it
can be used to estimate the size and age of the Universe. The Hubble constant indicates
the rate at which the Universe is currently expanding, so although it is called a ‘constant’ it
changes with time.
The Hubble constant is give by:
H0 =v/d
Where v is the radial velocity, d is the distance and the 0 denotes that the number derived
is the Hubble constant at the present time. The value for the Hubble constant is given in
kilometres per second per mega parsec (Mpc).
1. Now, calculate the Hubble Constant. Simply pick any point on the line and read off
the position along the y-axis and the x-axis for that point and then divide them as
shown above to find out the Hubble constant in terms of kilometres per second per
mega parsec. Add the value to your worksheet where indicated.
You’ve now successfully determined the rate of the expansion of the Universe!
Conclusion
Now that you have a value for the Hubble constant, one simple thing you can do to wrap
up the activity is estimate the age of the Universe. This can be done by finding the inverse
of the value of Ho
If the Universe has been expanding at a constant speed since its beginning, the Universe's
age would simply be 1/Ho
1. Calculate the age of the Universe using your value for the Hubble Constant.
2. Next, multiply you answer by 3.09 x 1019 km/Mpc to cancel the distance units. You
now have the age of the Universe in seconds!
3. Divide this number by the number of seconds in a year: 3.16 x 107 sec/yr
Now you have worked out the age of the Universe using real observational data!
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