Phys 321 Astronomy & Astrophysics II

Welcome to Phys 321 Astronomy & Astrophysics II

Course Instructor: Prof. Bin Chen Tiernan Hall 101 [email protected] NJIT Astronomy Courses

• The Physics Department has an undergraduate minor and a concentraon in Astronomy, which includes the following courses: § Phys 202, 202A – Intro to Astronomy and Cosmology § Phys 203, 203A – The Earth in Space § Phys 320 – Astronomy & Astrophysics I (last semester) § Phys 321 – Astronomy & Astrophysics II (this course) § Phys 322 – Observaonal Astronomy § Phys 420 – Special Relavity § Phys 421 – General Relavity § Phys 444 – Fluid and Plasma Dynamics Course Theme

• Use the physical laws and forces that govern our everyday lives § Gravity, electricity & magnesm… • To understand the cosmos • Use physics and math to quantavely determine the physical properes of astrophysical objects § Planets, stars, galaxies, clusters of galaxies, and the Universe Course Informaon: Material

• MW 11:30 am–12:55 pm. KUPF 208 • Textbook: “An Introducon to Modern Astrophysics”, 2nd Edion, by Carroll and Ostlie (the same book as Phys320) • Topics include stars, galaxies, and the Universe • Discussed in chapters 3, 5, 8, 9, 10, 12, 13, 15, 16, 17, 18, 24, 25, 26, 27, 28, and 29 of the textbook. Science and Math Disclaimer

• This is astrophysics course using quantave methods, NOT just general descripve astronomy • We use mostly freshman/sophomore physics (physics I, II, III) and calculus • We will also introduce some very basic quantum physics and relavity • Be prepared for quite some computaon in HWs, quizzes, and exams Course Informaon: Homework

• About one set a week, given Monday/Wednesday aer the lecture • Okay to work in small groups (<=2). But must be open and acknowledge who you have worked with. In all cases, you have to write answers clearly in your own words. • Due in a week (the following Monday/Wednesday) at the start of the class • Late submissions will have grades reduced by 50%. All assignments must be turned in by 4/30 to receive any credit. Class parcipaon: iClicker quizzes

q iClicker is required as part of the course

n Similar to requiring a textbook for the course n Can be purchased/rented at the NJIT bookstore n Can’t share with your classmate q iClicker use will be integrated into the course

n To be used during most or all lectures/discussions n iClicker questions will be worked into subject matter q Watch out for slides with an iClicker symbol q Each lecture will have some iClicker questions q You will receive both participation and performance credits, which constitute 20% of the final grade q We will start to use it from the second week (Wed 1/24) How will we use the i-Clicker?

• I post quesons on the slide during lecture.

• You answer using your i-Clicker remote.

• I can display a graph with the class results on the screen.

• We discuss the quesons and answers.

• You can get points (for parcipang and/or answering correctly)! These will be recorded (e.g., for quizzes and aendance). Class Informaon: Grading

• Homework (20%) • Class parcipaon and quizzes (20%) • Two in-class exams (15% each, 30% total) • Final Exam (30%) • Someme TBD in May 4–10 A >85 • Conversion chart to leer grade B+ >75-85 B >65-75 C+ >55-65 C 50-55 D, F < 50 Course Website: hps://web.njit.edu/~binchen/phys321

Phys 321: Lecture 1 Stars: Distances and Magnitudes

Prof. Bin Chen Tiernan Hall 101 [email protected] Cosmic Voyage: The Power of Ten The celesal sphere

19 How to measure distance

• Direct measurements § Using a standard ruler § Reflecon of laser light/waves • Indirect measurements § Angular size assuming a known physical size § Stereoscopic vision Review on Angle (Phys111) • What is the circumference S ? s s = (2π )r 2π = r • θ can be defined as the arc length r s s along a circle divided by the radius r: θ s θ = r • θ is a pure number, but commonly is given the arficial unit, radian (“rad”) q Whenever using rotaonal equaons, you MUST use angles expressed in radians Conversions between radians and degrees qComparing degrees and radians

2π (rad) = 360! π (rad) =180! qConverng from degrees to radians π α rad = α degrees ( ) 180° ( ) qConverng from radians to degrees

180! 360° α(degrees) = α(rad) 1rad = = 57.3° π 2π Unit Basicconversion Basic Units Units of ofAngular Angular MeasurementMeasurement in inAstronomy Astronomy !• Degree,Degree, arc-minute, and arc-second arc-minute and arc-second ! Degree, arc-minute and arc-second ! 1 (deg! ) = 60'(arcmin) 1 (deg) = 60'(arcmin) 1'(arcmin) = 60"(arcsec) 1'(arcmin) = 60"(arcsec) ! 1 (deg! ) = 3600"(arcsec) 1 (deg) = 3600"(arcsec) ! Converting from radians to degrees • From radians to arc-second ! Converting from radians to degrees ! 180 ! ! 1 rad 18(r0ad) 57.2958 3!438' 206265" 1 r=ad = (ra=d) = 57.29=58 = 34=38' = 206265" π π

October 28 - NovemberOctober 3 28 - November 3 2.4 Angular Distance

! If you draw lines from your eye to each of two stars, the angle between the lines is the angular distance between the two stars. ! Note: here we refer to the distance projected to the surface of an imaginary celestial sphere centered at the observer, as if the two objects were in this same spherical surface.

" d % Q: does the radius of this δ" ≈ 206, 265$ ' Angular Size and Angular #Distance D & sphere matter?

Use in Astronomy

d • Angular size of an object δ ≈ α declines with increasingD physical distance for an d " d % object of a ﬁxed physical size δ" ≈ 206, 265$ ' s # D &

! δ depends upon the actual diameter of the object d ! δ also depends upon the actual diameter of the object D ! Same object: the further away, the smaller? δ ~ 1/D ! Different objects of the same size, the further away, the larger? δ ~ d/D ! Different objects with different sizes may have the same angular sizes? Measuring angular distances in the sky Calculang angular size in astronomy

δ d tan( ) = For small angles, tanα ≈ α, so 2 2D tanα = α α ≈ s / D Angular size ≈ physical size / distance Angular size of the Sun

• What is the angular size of the Sun in arcsec? • The Sun’s radius is 6.96 x 105 km 2.3 Angular• Sun-Earth distance is 1.50 x 10 Size in Astronomy 8 km

! d $ δ = 2arctan# & " 2D % when D >> d

d δ ≈ D ! δ depends upon the actual diameter of the object d ! δ also depends upon the actual diameter of the object D ! δ is the angular diameter in unit of radian " d % " d % δ" ≈ 206, 265$ ' δ ' ≈ 3, 438$ ' # D & # D & Angular size of planets

Angular size in arcsec Relave Size (orange bar is the maximum value for Venus) Venus Jupiter

Saturn

Mars Mercury Uranus Neptune

Small angle approximaon works prey well for all celesal bodies! How to measure astronomical distance? § Angular size assuming a known physical size

Angular size ≈ physical size / distance α ≈ s / D

Distance ≈ angular size * physical size D ≈ αs § Stereoscopic vision Stereoscopic Vision: 3D movie Stereoscopic Vision: 3D movie

From reallusion.com Parallax: Stereoscopic in astronomy Measuring distances using parallax

Annual Parallax: observingp is the parallax angle in a celestial object 6 monthsradians apart,1AU B becomes1 the Sun-Earth distance, i.e., 1 AU. d = ≈ AU AU is the Astronomical Unit, which is Parsectan (Parp allaxp arcsecthe mean distance from the Sun to ), or pc, is an astronomical distance units. 1 pc isEarth: 1.496 x 10 the distance of8 km an object whose parallax is 1 arcsec; 1pc = 3.26 ly.

9/22/15 Q: what is the parallax of Proxima Centauri? The Continuous Spectrum of Light The Continuous Spectrum of Light

d p 2B d p 2B

FIGUREFIGURE 1 TrigonometricTrigonometric parallax: parallax:d d B/B/tantanp. p. ==

EarthEarth

p p SunSun dd

FIGUREParallax 2 Stellar parallax: connued d 1/p pc. FIGURE 2 Stellar parallax: d 1/p′′ ′′ pc. == 1AU 1 where the small-angle approximationd = tan p ≈p hasAU beenp employed is the parallax angle in for the parallaxradians angle where the small-angle approximation tantanpp≃ pp has been employed for the parallax angle p pmeasuredmeasured in inradiansradians.. Using Using 1 radian 5757.≃2957795.2957795◦ 206264206264.806.806′′ to convertto convertp toppto′′ p in units of arcseconds produces = ◦= ′′ ′′ in units of arcseconds produces Converng to = arcseconds=, knowing 1 radian ≈ 206,265” 206,265 d 206,265AU. d ≃ p′′ AU. P” is the parallax angle in arcsecs ≃ p′′ Defining a new unit of distance,Defining a new unit of distance, the the parsec (parallax-second, abbreviatedparsec (par pc),allax- assec 1ond, or pc Defining a new unit5 of distance, the parsec16 (parallax-second, abbreviated pc), as 1= pc 2.06264806 10 AU 3.0856776pc), as 1 pc ≈ 206,265 AU ≈ 3.086 x 1010 m leads to 16 m, which leads to 2.06264806 ×105 AU = 3.0856776× 1016 m leads to = × = × • 1 pc ≈ 3.262 light year 1 d pc. • Measured parallax angle of 1” gives a (1) 1 distance of 1 pc = p′′ d pc. • Smaller parallax angle -> larger distance (1) = p′′ By definition, when the parallax angle p 1 , the distance to the star is 1 pc. Thus 1 parsec = ′′ Byis definition, the distance when from the which parallax the radius angle ofp Earth’s1 orbit,, the distance 1AU, subtends to the star an angle is 1 pc. of 1 Thus′′.Another 1 parsec = ′′ isunit the distance of distance from often which encountered the radius is of the Earth’slight-year orbit,(abbreviated 1AU, subtends ly), the an angle distance of 1 traveled′′.Another by light through a vacuum in one Julian year: 1 ly 9.460730472 1015 m. One parsec unit of distance often encountered is the light-year=(abbreviated ly),× the distance traveled byis light equivalent through to 3.2615638 a vacuum ly.in one Julian year: 1 ly 9.460730472 1015 m. One parsec is equivalent to 3.2615638 ly. = ×

The Continuous Spectrum of Light

d p 2B

FIGURE 1 Trigonometric parallax: d B/tan p. =

Earth

p Sun d

FIGURE 2 Stellar parallax: d 1/p′′ pc. = where the small-angle approximation tan p p has been employed for the parallax angle ≃ p measured in radians. Using 1 radian 57.2957795◦ 206264.806′′ to convert p to p′′ in units of arcseconds produces = =

Proxima206,265 Centauri: the nearest star d AU. ≃ p′′ • The measured parallax angle of Proxima Deﬁning a new unit of distance,Centauri is 0.77”, what is its distance in the parsec (parallax-second, abbreviated pc), as 1 pc 2.06264806 105 AU 3.0856776 1016 m leads to = × = pc? And light years? × 1 d pc. d ≈ 1.3 pc ≈ 4.23 ly (1) = p′′

• Proxima Centauri has a planet (called By deﬁnition, when the parallax angle p 1′′, the distance to the star is 1 pc. Thus 1 parsec Proxima= b) located in the habitable zone is the distance from which the radius of Earth’s orbit, 1AU, subtends an angle of 1′′.Another unit of distance often encounteredof the star. Stephen Hawking has a plan is the light-year (abbreviated ly), the distance traveled by light through a vacuum in oneto build a solar sail (called “Breakthrough Julian year: 1 ly 9.460730472 1015 m. One parsec = × is equivalent to 3.2615638 ly. Starshot”) to travel to the stellar system. Image of Proxima Centauri by the In theory, the nanocra can reach 20% of Hubble Space Telescope the speed of light. If so, how long does it take to reach the star?

The Continuous Spectrum of Light

Even Proxima Centauri, the nearest star other than the Sun, has a parallax angle of less than 1′′. (Proxima Centauri is a member of the triple star system α Centauri, and has a parallax angle of 0.77′′. IfThe Earth’s Continuous orbit Spectrum around of Light the Sun were represented by a dime, then Proxima Centauri would be located 2.4 km away!) In fact, this cyclic change in a star’s Even Proxima Centauri, the nearest star other than the Sun, has a parallax angle of less position is so difficultthan 1′′. (Proxima to detect Centauri that is a it member was of not the triple until star 1838 system thatα Centauri, it was and has first a measured, by Friedrich Wilhelmparallax Bessel angle (1784–1846), of 0.77′′. If Earth’s orbit a German around the Sun mathematician were represented by and a dime, astronomer. then 1 Proxima CentauriThe would Continuous be located Spectrum 2.4 km of Light away!) In fact, this cyclic change in a star’s positionParallax is so difficult Example to detect that it was not until 1838 that it was first measured, by 1 FriedrichEven Wilhelm Proxima Bessel Centauri, (1784–1846), the nearest a star German other thanmathematician the Sun, has and a parallax astronomer. angle of less than 1′′. (Proxima Centauri is a member of the triple star system α Centauri, and has a Example 1.1. Inparallax 1838, angle after of 0.77 4′′. years If Earth’s of orbit observing around the Sun 61 were Cygni, represented Bessel by a dime, announced then his mea- ExampleProxima 1.1. CentauriIn 1838, would after be located 4 years 2.4 of observing km away!) 61 In Cygni, fact, this Bessel cyclic announced change in his a star’s mea- surement of a parallaxsurementposition angle of is a parallax so difficult of 0 angle.316 to detect of′′ 0.316for that′′ for it that was that not star. star. until This This 1838 corresponds that corresponds it was to afirst distance measured, toof a by distance of Friedrich Wilhelm Bessel (1784–1846), a German mathematician and astronomer.1 1 1 1 d pc 1 pc 3.16 pc 10.3 ly, Exampled 1.1. Inpc 1838,= p′′ after= 4 years0.316pc of observing= 3.16 61 Cygni,= pc Bessel10 announced.3 ly, his mea- surement of a parallax angle of 0.316′′ for that star. This corresponds to a distance of within 10%= of thep′′ modern= value0. 3.48316 pc. 61 Cygni= is one of the= Sun’s nearest neighbors. 1 1 d pc pc 3.16 pc 10.3 ly, = p′′ = 0.316 = = within 10% of the modernFrom 1989 to value 1993, the 3.48 European pc. Space 61 CygniAgency’s (ESA’s) is one Hipparcos of the Space Sun’s Astrometry nearest neighbors. 2 Missionwithin operated 10% of the high modern above value Earth’s 3.48 distorting pc. 61 Cygni atmosphere. is one of theThe Sun’s spacecraft nearest neighbors.was able to measure parallax angles with accuracies approaching 0.001′′ for over 118,000 stars, cor- responding to a distance of 1000 pc 1 kpc (kiloparsec). Along with the high-precision HipparcosFrom experiment 1989 to 1993, aboard the European the spacecraft,≡ Space Agency’s the lower-precision (ESA’s) Hipparcos Tycho Space experiment Astrometry pro- From 1989 to 1993,Mission the operated European high above Space Earth’s distorting Agency’s atmosphere. (ESA’s)2 The spacecraft Hipparcos was able Space to Astrometry duced a catalog of more than 1 million stars with parallaxes down to 0.02′′ –0.03′′. The measure parallax angles with accuracies approaching 0.001′′ for over2 118,000 stars, cor- Mission operatedtwo highresponding catalogs above were to a published distance Earth’s of in 1000 1997 distorting pc and1 are kpc available (kiloparsec). atmosphere. on CD-ROMs Along with andThe the the high-precision World spacecraft Wide was able to Web.Hipparcos Despite theexperiment impressive aboard accuracy the spacecraft,≡ of the Hipparcos the lower-precision mission, the Tycho distances experiment that were pro- measure parallax angles with accuracies approaching 0.001′′ for over 118,000 stars, cor- obtainedduced are a catalog still quite of more small than compared 1 million to the stars 8-kpc with distance parallaxes to the down center to 0 of.02 our′′ –0 Milky.03′′. Way The responding to a distanceGalaxy,two catalogs so stellar of were 1000 trigonometric published pc in parallax 19971 kpc and is are currently (kiloparsec). available useful on CD-ROMs only Alongfor andsurveying the withWorld the Wide local the high-precision neighborhoodWeb. Despite of the the Sun. impressive≡ accuracy of the Hipparcos mission, the distances that were Hipparcos experimentHowever,obtained aboard are within still quite the the next small spacecraft, decade, compared NASA to the plans the 8-kpc to lower-precision distance launch the to the Space center Interferometry of our Tycho Milky WayMis- experiment pro- sionGalaxy, (SIM PlanetQuest). so stellar trigonometric This observatory parallax will is currently be able useful to determine only for positions, surveying distances, the local duced a catalog of moreneighborhood than of the 1 Sun.million stars with parallaxes down to 0.02′′ –0.03′′. The and proper motions of stars with parallax angles as small as 4 microarcseconds (0.000004′′), two catalogs wereleading publishedHowever, to the direct within in determination 1997 the next decade, and of distances are NASA available plans of objects to launch up to on the 250 Space CD-ROMs kpc away, Interferometry assuming and Mis- that the World Wide the objectssion (SIM are PlanetQuest). bright enough. This In addition, observatory ESA will will be launch able to the determineGaia mission positions, within distances, the next Web. Despite the impressiveand proper motions accuracy of stars with parallax of the angles Hipparcos as small as 4 micro mission,arcseconds (0 the.000004 distances′′), that were decadeleading as well, to the which direct will determination catalog the of brightest distances 1 of billion objects stars up to with 250 parallax kpc away, angles assuming as small that obtained are still quiteas 10the microarcseconds. objects small are compared bright With enough. the In anticipated to addition, the ESA 8-kpc levels will of launch distanceaccuracy, the Gaia these tomission missions the within center will the be next able of our Milky Way to catalogdecade stars as well, and which other objectswill catalog across the the brightest Milky 1 Way billion Galaxy stars andwith even parallax in nearby angles galaxies. as small Galaxy, so stellarClearly trigonometricas 10 these microarcseconds. ambitious projects parallax With the will anticipated provide is currently an levels amazing of accuracy, wealth useful these of new missions only information for will be surveying about able the local neighborhood of thethe three-dimensionalto Sun. catalog stars and otherstructure objects of our across Galaxy the Milky and the Way nature Galaxy of and its constituents.even in nearby galaxies. Clearly these ambitious projects will provide an amazing wealth of new information about However, withinthe the three-dimensional next decade, structure NASA of our Galaxy plans and tothe nature launch of its constituents.the Space Interferometry Mis- sion (SIM PlanetQuest). This observatory will be able to determine positions, distances, 1Tycho Brahe had searched for stellar parallax 250 years earlier, but his instruments were too imprecise to find it. Tycho concluded that Earth does not move through space, and he was thus unable to accept Copernicus’s model and proper motions of1 stars with parallax angles as small as 4 microarcseconds (0.000004′′), of a heliocentricTycho Brahe Solar had searched System. for stellar parallax 250 years earlier, but his instruments were too imprecise to find it. leading to the direct2Astrometry determinationTycho concludedis the subdiscipline that Earth does of of astronomy not distances move through that is concerned space, of and objects withhe was the thus three-dimensional unable up to to accept 250 positions Copernicus’s kpc of celestial away, model assuming that objects.of a heliocentric Solar System. 2Astrometry is the subdiscipline of astronomy that is concerned with the three-dimensional positions of celestial the objects are brightobjects. enough. In addition, ESA will launch the Gaia mission within the next decade as well, which will catalog the brightest 1 billion stars with parallax angles as small as 10 microarcseconds. With the anticipated levels of accuracy, these missions will be able to catalog stars and other objects across the Milky Way Galaxy and even in nearby galaxies. Clearly these ambitious projects will provide an amazing wealth of new information about the three-dimensional structure of our Galaxy and the nature of its constituents.

1Tycho Brahe had searched for stellar parallax 250 years earlier, but his instruments were too imprecise to ﬁnd it. Tycho concluded that Earth does not move through space, and he was thus unable to accept Copernicus’s model of a heliocentric Solar System. 2Astrometry is the subdiscipline of astronomy that is concerned with the three-dimensional positions of celestial objects.

Topic 2: The Magnitude Scale Apparent magnitude

• Some stars appear brighter and some stars appear fainter. How to characterize this “brightness”? • Apparent magnitude, introduced originally by Greek astronomer Hipparchus (190-120 BC) • Magnitudes 1 through 6 for the brightest to the dimmest stars visible to the naked eye • The brighter the object, the smaller the magnitude • The magnitude scheme follows a logarithmic scale • A m=1 star is 100 x brighter than a m=6 star • A m=1 star is 1001/5 brighter than a m=2 star • A m=1 star is 1002/5 brighter than a m=3 star • A m=1 star is 1003/5 brighter than a m=4 star • … Redeﬁne “brightness”

• The apparent “brightness” of an object is actually measured in terms of the radiant ﬂux F received from the object

• That is, brightness is really the radiant energy passing through a given area in a given me. For our eye, it is the area of the pupil.

• Unit: J/s/m2, or W/m2 Inverse Square Law

• The “brightness”, or ﬂux decreases at greater distance following the inverse square law: 1 The amount of luminosity F ∝ passing through each d 2 sphere is the same.

Why? Area of sphere: Energy is conserved! 4π (radius)2 The total amount of power passing through each sphere is the same Divide luminosity by area to get Flux, or “apparent brightness”. Area of sphere: 4π(distance)2

The Continuous Spectrum of Light

Imagine a star of luminosity L surrounded by a huge spherical shell of radius r. Then, assuming that no light is absorbed during its journey out to the shell, the radiant ﬂux, F , measured at distance r is related to the star’s luminosity by

L LuminosityF , and Flux (2) = 4πr2 the denominator being simply the area of the sphere. Since L does not depend on r, the • The intrinsic “power” of the star is defined as the radiant flux is inversely proportional to the square of the distance from the star. This is the luminosity L, which is the 4 total energy given off by well-known inverse squarethe star law forper unit me light. (unit: J/s, or W)

Example 2.1. The luminosity ofFor our Sun: the Sun is L 3.839 1026 W. At a distance of 1 AU 1.496 1011 m, Earth receives a radiant flux⊙ = above its× absorbing atmosphere of = × Luminosity • So, the flux is L Flux = 2 F 1365 W4π(distance) m− . 2 = 4πr2 = This value of the solar flux is known as the solar irradianceL , sometimes also called the F = 6 solar constant. At a distance of 10 pc 2.063 410dAU,2 an observer would measure the radiant flux to be only 1/(2.063 106=)2 as large.× π That is, the radiant flux from the Sun 10 2 × would be 3.208 10 Wm− at a distance of 10 pc. × −

Absolute Magnitude Using the inverse square law, astronomers can assign an absolute magnitude, M, to each star. This is deﬁned to be the apparent magnitude a star would have if it were located at a distance of 10 pc. Recall that a difference of 5 magnitudes between the apparent magnitudes of two stars corresponds to the smaller-magnitude star being 100 times brighter than the larger-magnitude star. This allows us to specify their ﬂux ratio as

F 2 (m1 m2)/5 100 − . (3) F1 =

Taking the logarithm of both sides leads to the alternative form:

F1 m1 m2 2.5 log . (4) − = − 10 F ! 2 " The Distance Modulus The connection between a star’s apparent and absolute magnitudes and its distance may be found by combining Eqs. (2) and (3):

2 (m M)/5 F10 d 100 − , = F = 10 pc ! " 4If the star is moving with a speed near that of light, the inverse square law must be modiﬁed slightly.

The ContinuousThe Continuous Spectrum Spectrum of Light of Light

ImagineImagine a star a of star luminosity of luminosityL surroundedL surrounded by a by huge a huge spherical spherical shell shell of radius of radiusr. Then,r. Then, assumingassuming that no that light no light is absorbed is absorbed during during its journey its journey out to out the to shell, the shell, the radiant the radiant ﬂux, ﬂux,F , F , measuredmeasured at distance at distancer is relatedr is related to the to star’s the star’s luminosity luminosity by by

L F ,L (2) F 2 , (2) = 4π=r 4πr2 the denominatorthe denominator being being simply simply the area the area of the of sphere. the sphere. Since SinceL doesL does not depend not depend on r on, ther, the radiant ﬂux is inversely proportional to the square of the distance from the star. This is the radiant ﬂux is inversely proportional4 to the square of the distance from the star. This is the well-knownwell-knowninverseinverse square square law for law light.for light.4

26 Example 2.1. The luminosity of the Sun is L 3.839 10 W.26 At a distance of Example 2.1. 11 The luminosity of the Sun⊙ is=L 3.×839 10 W. At a distance of 1 AU 1.496 10 m,11 Earth receives a radiant flux above⊙ = its absorbing× atmosphere of 1= AU 1×.496 10 m, Earth receives a radiant flux above its absorbing atmosphere of = × L 2 F L 1365 W m− . 2 =F4πr2 = 1365 W m− . = 4πr2 = This value of the solar flux is known as the solar irradiance, sometimes also called the This value of the solar flux is known as the solar6 irradiance, sometimes also called the solar constant. At a distance of 10 pc 2.063 10 AU,6 an observer would measure the solar constant. At a distance of 106= pc2 2.×063 10 AU, an observer would measure the radiant flux to be only 1/(2.063 10 ) as6=2 large. That× is, the radiant flux from the Sun radiantBrightness flux to be10 only 1 /(and22.×063 Apparent10 ) as large. That Magnitude is, the radiant flux fromm the Sun would be 3.208 10− Wm10 − at a2 distance× of 10 pc. would be 3×.208 10 Wm− at a distance of 10 pc. × − Absolute Magnitude Absolute• Recall: A difference of 5 magnitudes, or m Magnitude 1-m2 = 5, Using the inversecorresponds to the square law, astronomerssmaller-magnitude can assign an absolutestar (with m magnitude, M, to) each star. ThisUsing is the defined inverse to be square the apparent law, astronomers magnitude can a assign star would an absolute have if magnitudeit were located,2M, at to a each distancestar. of This100 mes brighter 10 pc. is defined Recall that to be a difference the apparent in its apparent brightness than the of 5 magnitude magnitudes a star between would the have apparentif it were magnitudes located at a distance of 10 pc. Recall that a difference of 5 magnitudes between the apparent magnitudes of two starslarger-magnitude star (with m corresponds to the smaller-magnitude star1), or F being2 100/F1 times=100 brighter than the larger-magnitudeof two stars star. corresponds This allows to the us smaller-magnitude to specify their flux star ratio being as 100 times brighter than the larger-magnitude• It means: star. This allows us to specify their flux ratio as F 2 (m1 m2)/5 F100 − . (3) 2 (m1 m2)/5 F1 = 100 − . (3) F1 = Taking the• logarithm of both sides leads to the alternative form: TakingTaking the logarithm of both sides the logarithm of both sides leads to the alternative form: F1 m1 m2 2.5 log . (4) − = − 10 F F1 m1 m2 2.5 log 2 . (4) − = − ! 10 "F ! 2 " The Distance Modulus The Distance Modulus The connection between a star’s apparent and absolute magnitudes and its distance may be foundThe by combiningconnection Eqs. between (2) and a star’s (3): apparent and absolute magnitudes and its distance may be found by combining Eqs. (2) and (3): 2 (m M)/5 F10 d 100 − , 2 (m M)/= 5F =F10 10 pc d 100 − ! " , 4 = F = 10 pc If the star is moving with a speed near that of light, the inverse square! law must" be modified slightly. 4If the star is moving with a speed near that of light, the inverse square law must be modified slightly.

The Continuous Spectrum of Light

L F , (2) = 4πr2 the denominator being simply the area of the sphere. Since L does not depend on r, the radiant ﬂux is inversely proportional to the square of the distance from the star. This is the well-known inverse square law for light.4

Example 2.1. The luminosity of the Sun is L 3.839 1026 W. At a distance of The Continuous Spectrum of Light 1 AU 1.496 1011 m, Earth receives a radiant flux⊙ = above its× absorbing atmosphere of = × Imagine a star of luminosity L surrounded by a huge spherical shell of radius r. Then, assuming that no lightL is absorbed during its journey2 out to the shell, the radiant flux, F , F 1365 W m− . measured at distance= 4πr isr2 related= to the star’s luminosity by This value of the solar flux is known as the solar irradianceL , sometimes also called the F , (2) solar constant. At a distance of 10 pc 2.063 106 =AU,4πr an2 observer would measure the 6=2 × radiant flux to be onlythe 1 denominator/(2.063 being10 ) simplyas large. the area That of the is, sphere. the radiantSince L does flux not from depend the on Sunr, the 10 2 × would be 3.208 10−radiantWm flux− is inverselyat a distance proportional of 10 to pc. the square of the distance from the star. This is the × well-known inverse square law for light.4

Absolute MagnitudeExample 2.1. The luminosity of the Sun is L 3.839 1026 W. At a distance of 1 AU 1.496 1011 m, Earth receives a radiant flux⊙ = above its× absorbing atmosphere of Using the inverse square law,= astronomers× can assign an absolute magnitude, M, to each L star. This is defined to be the apparent magnitudeF a star would1365 W have m 2. if it were located at a 2 − distance of 10 pc. Recall that a difference of 5 magnitudes= 4πr = between the apparent magnitudes of two stars correspondsThis value to the of smaller-magnitude the solar flux is known as star the beingsolar irradiance 100 times, sometimes brighter also than called the the solar constant. At a distance of 10 pc 2.063 106 AU, an observer would measure the larger-magnitude star.radiant This flux allows to be us only to 1 specify/(2.063 their106=)2 fluxas large. ratio× That as is, the radiant flux from the Sun 10 2 × would be 3.208 10 Wm− at a distance of 10 pc. × − F 2 (m1 m2)/5 Absolute Magnitude 100 − . (3) F1 = UsingAbsolute the inverse Magnitude square law, astronomers M can assign an absolute magnitude, M, to each star. This is defined to be the apparent magnitude a star would have if it were located at a Taking the logarithm•distance ofApparent magnitude m depends strongly on both of sides 10 pc. leadsRecall that to the a difference alternative of 5 magnitudes form: between the apparent magnitudes ofdistance – inconvenient to compare the intrinsic two stars corresponds to the smaller-magnitude star being 100 times brighter than the larger-magnitudebrightness among stars star. This allows us to specifyF1 their flux ratio as m1 m2 2.5 log10 . (4) −L = − F F2 2 ! (m"1 m2)/5 F = 100 − . (3) 2 F1 = The Distance Modulus 4πd Taking the logarithm of both sides leads to the alternative form: The connection between• We define a star’sAbsolute Magnitude apparent and absolute (M) of a star as the magnitudes and its distance may be apparent magnitude the star would have if it were F1 found by combining Eqs. (2) and (3): m1 m2 2.5 log . (4) placed at a distance of 10 pc. − = − 10 F ! 2 " 2 The Distance(m ModulusM)/5 F10 d 100 − , = F = 10 pc The connection between a star’s apparent! and absolute" magnitudes and its distance may be found by combining Eqs. (2) and (3): 4If the star is moving with a speed near that of light, the inverse square law must be modified slightly. 2 (m M)/5 F10 d 100 − , = F = 10 pc ! " 4If the star is moving with a speed near that of light, the inverse square law must be modified slightly.

The Continuous Spectrum of Light

Example 2.1. The luminosity of the Sun is L 3.839 1026 W. At a distance of 1 AU 1.496 1011 m, Earth receives a radiant flux⊙ = above its× absorbing atmosphere of = × L 2 F 1365 W m− . = 4πr2 = This value of the solar flux is known as the solar irradiance, sometimes also called the solar constant. At a distance of 10 pc 2.063 106 AU, an observer would measure the radiant flux to be only 1/(2.063 106=)2 as large.× That is, the radiant flux from the Sun 10 2 × would be 3.208 10 Wm− at a distance of 10 pc. × −

F 2 (m1 m2)/5 100 − . (3) F1 =

Taking the logarithm of both sides leads to the alternative form:

F1 m1 m2 2.5 log . (4) − = − 10 F ! 2 " The Distance ModulusDistance Modulus The connection between a star’s apparent and absolute magnitudes and its distance may be found by combining• From the expression of Absolute Magnitude Eqs. (2) and (3):

2 (m M)/5 F10 d 100 − , = F = 10 pc ! " 4If the star is moving with• Take logarithm on both sides a speed near that of light, the inverse square law must be modiﬁed slightly. ⎛ d ⎞ m − M = 5log10 ⎜ ⎟ ⎝10pc ⎠ • The diﬀerence is a measure of the distance, called the m − M distance modulus • Very useful in measuring distance if m is measured and M can be inferred (standard candles) –> this principle is the key in the discovery of the accelerang expansion of the Universe – 2011 Nobel Prize Summary

• Distance in astronomy

• Parallax

• Apparent magnitude

• Flux, Luminosity, and inverse square law

• Absolute magnitude and distance

• Step 1: Switch your frequency to “AB” üPress and hold power buon unl the indicator starts to ﬂash üEnter AB using the buons (ABCD) on the clicker

• Step 2: We will start a Roll Call session to register your iClickers. You will see your name on the screen. Enter the 2-leer codes associated with your name. Clicker Quesons Two stars in a binary system are observed to orbit each other with a ﬁxed angular separaon 1”. If the system has a measured parallax angle of p=0.1”, the physical separaon between the two stars is:

A. 0.1 AU B. 1 Au C. 0.01 AU D. 10 AU E. 100 AU Two stars A and B have a luminosity rao of LA/LB = 64. However an observer claim that they appear to have the same brightness. From parallax measurements, people ﬁnd that star A is located at 80 pc. How far is star B in pc?

A. 8 B. 10 C. 1.25 D. 5120 E. 640