rotasi galaksi 2011

Kuliah Fisika Galaksi

5 Mei 2010

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Populasi Bintang di Galaksi Bima Sakti
Schematic (edge-on) view of the major components Source: Roland Buser http://www.astro.unibas.ch/forschung/rb/structure.shtml) The disk and halo structure perpendicular to the Galactic plane near the solar neighborhood
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Gerak Bintang di Galaksi
Bintang2 di Galaksi merupakan anggota dari komponen galaksi yang berbeda2, perbedaannya tidak hanya dlm distribusi ruang saja, tetapi juga kinematikanya.Gerak yang mendominasi bintang2 dan gas di piringan galaksi adalah rotasi terhadap pusat galaksi dg orbit berbentuk lingkaran.Bintang2 di piringan tebal (thick disk) berotasi lebih lambat daripada yang berada di piringan tipis (thin disk). Gerak acak (random motion) bintang tersebut lebih besar.Rotasi bintang2 di halo tidak seperti yang ada di piringan, gerak acak mereka lebih besar dan orbitnya berbentuk elips.
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Pertanyaan :
Bagaimana kita tahu tentang gerak bintang di galaksi ?Bagaimana menentukan kecepatan rotasi di piringan ?Bagaimana kita dapat menentukan massa Galaksi ?
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Kerangka Acuan
Untuk mempelajari dinamika galaksi, kerangka acuan dasar pada galaksi sangat diperlukan. Kecepatan bintang pada kerangka acuan ini sering diberikan dalam koordinat silinder (P,Q,Z) atau (VR, Vf, VZ)
P : sepanjang arah radial pd bidang galaksi, positif ke arah luar (anti-center), l=180, b=0

W: arah tangential pd bidang galaksi,positif ke arah rotasi galaksi, l=90, b=0

Z: arah tegaklurus bidang galaksi, positif ke arah utara, b=90

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Local Standard of Rest (LSR)
Kita definisikan sebuah kerangka acuan pd bidang galaksi yg bergerak dalam orbit lingkaran mengelilingi pusat galaksi sebagai standar diam lokal (LSR)LSR adalah kerangka acuan lokal yg terletak di daerah sekitar matahari yg bergerak dlm orbit lingkaranSebuah bintang yg bergerak dlm orbit lingkaran pd bidang galaksi akan tetap pada geraknya karena :Galaksi berbentuk simetri sumbu, F=F(R,Z)Simetri thd bidang galaksiDalam keadaan steady state
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Local Standard of Rest (LSR)
Daerah sekitar matahari (Solar Neighborhood, SN) didefinisikan sebagai ruang bola yg berukuran kecil (thd galaksi) yg berpusat di matahari dan terdiri dari sample suatu tipe bintangDisk : SN adalah sebuah bola dg radius 50-100 pc (1% dari piringan galaksi)Halo : radius ~ 1 kpc (1% dari halo)Kecepatan matahari
terhadap LSR

(u,v,w) ~ (-10,5,7) km/s

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31.pdf
Wednesday Extragalactic astronomy tools

Multi-wavelength astronomy Starcounts Extinction Globular Clusters

Milky way components: Disk: exponential profile, thin/thick disk Stellar Halo, Pop I/II stars, Globular Clusters Age-Metallicity Relation ISM - HII (Hot Ionized), HI (atomic), H2 (molecular) Bulge Dark Matter Halo Satellite galaxies

Kinematics of the Galaxy I - Rotation

CurvesJL 1.3,1.6; Carroll & Ostlie pp 898-907, 915-933

Today Disk, Halo Motions Galactic Coordinates Measuring Distance: parallax pulsating stars Rotation Curves Local Standard of Rest Rotation speed from halo star motions Tangent-Point Method and Rotation Curves Weighing in New results?

Chung et al:http://www.astro.columbia.edu/~archung/research/CRdisk/crdisk.html

Other Galaxies Motions

Radial velocities can be measured with spectra

Can get rotation curves, (radial) velocity dispersions
http://www.astro.columbia.edu/~archung/research/CRdisk/crdisk.htmlhttp://www.astro.columbia.edu/~archung/research/CRdisk/crdisk.html
Where are we and how fast

are we moving? Within our own galaxy, more

difficult! Even though we know were

edge on, even local position and rotation speed are uncertain

The Open University

30 kpc

6 kpc

1

50 kpc

dark-matter haloand stellar halo

bulge

(a)

(b)

stellar disc

bulge

stellardisc

locationof Sun

locationof Sun

1 kpc

gaseousdisc


JL Fig 1.5

Even now, numbers

jumping around Recent (announced Tues at

AAS meeting) found evidence of ~15% higher rotation rate than generally understood

Using parallax of very bright microwave objects (masers)

Rotation curves of other spirals

Rotation curves of spiral galaxies quite generally have flat rotation curves

Non-visible matter contributing to mass of galaxies, not to light

Dark Matter

Sofue et al (1997)http://www.ioa.s.u-tokyo.ac.jp/~sofue/rotation/fig1/all-new.gif
http://www.ioa.s.u-tokyo.ac.jp/~sofue/rotation/fig1/all-new.gifhttp://www.ioa.s.u-tokyo.ac.jp/~sofue/rotation/fig1/all-new.gifhttp://www.ioa.s.u-tokyo.ac.jp/~sofue/rotation/fig1/all-new.gifhttp://www.ioa.s.u-tokyo.ac.jp/~sofue/rotation/fig1/all-new.gif
M

v

r

r

vr

8

Circular velocity around

a mass Assuming symmetry, constant

circular orbit around a (spherical) mass distribution only depends on total mass enclosed:

v2r(R) =GM(< R)

r

`Solid Body

vr = const = vr/r r1

Flat Rotation curve

Keplerian

vr r1/2

r3/2

Examples of

differential rotation

= constvr = r r

Motion w/rt Galactic Centre

Sgr A* shows very regular proper motion, even though surrounded by thousands of starsMotions entirely consistent with being due only to solar motion

Reid & Brunthaler (2004)http://www.journals.uchicago.edu/doi/abs/10.1086/424960

v! = (0.4 0.9)km s1
http://www.journals.uchicago.edu/doi/abs/10.1086/424960http://www.journals.uchicago.edu/doi/abs/10.1086/424960

But dont know distance from galactic centre; cant automatically turn this into a rotational velocity

RA & Dec for Solar System

ESO: http://www.eso.org/public/outreach/eduoff/vt-2004/Background/Infol2/EIS-D3.html
http://www.eso.org/public/outreach/eduoff/vt-2004/Background/Infol2/EIS-D3.htmlhttp://www.eso.org/public/outreach/eduoff/vt-2004/Background/Infol2/EIS-D3.html
Galactic Coordinates

CO Fig 24.17


CO Fig 24.19

VU

Galactic Orbits

Stars in disk typically have aligned circular orbits

In halo, orbits fill halo

PopI vs PopIIJoshua Barnes,

http://www.ifa.hawaii.edu/~barnes/ast110_01/
http://www.ifa.hawaii.edu/~barnes/ast110_01/tmwaog/orbits.gifhttp://www.ifa.hawaii.edu/~barnes/ast110_01/tmwaog/orbits.gif
Parallax

1 AUp

D/tan(p) ~ D/p

Definition: 1pc = (1 AU)/(1 (in radians))

But measuring them is hard

First parallax measurement: Bessel (1939)

61 Cygni

Measured parallax of 0.31360.015 (d = 3.04-3.35 pc)

Current distance: 3.496 0.007 pc

Hipparcos mission

For stars larger than few parsec away, limits on seeing prevent good measurements

118,000 stars with high-precision distances, proper motions

Still limited by baseline - ~1 AU Hipparcos: 1989-1993

http://www.rssd.esa.int/index.php?project=HIPPARCOS
http://www.rssd.esa.int/index.php?project=HIPPARCOShttp://www.rssd.esa.int/index.php?project=HIPPARCOS
Photometric Distances

Once know stars distances in local neighborhood, know real luminosity (L) and absolute magnitude M

From observed m, can get distance modulus and distance

But extinction! m is less than it would be with just distance effect.

d = (10pc)10(mM)/5

But extinction can be characterized

Depends on wavelength

So by finding out how much the difference is in different wavelengths, can estimate the magnitude of effect

So can still get a distance

Alves, Lada & Lada (2001)

Visible

Near IR

d = (10pc)10(mMA)/5


Sketch of how extinction behaves

Less effective at low frequencies

Some similarities to scattering of light in atmosphere

(eg, why is sky blue?)http://www.jb.man.ac.uk/distance/life/sample/stars/index.html

Jodrell Bank Observatory
http://www.jb.man.ac.uk/distance/life/sample/stars/index.htmlhttp://www.jb.man.ac.uk/distance/life/sample/stars/index.html
Spectroscopic Binaries

R. Poggehttp://www-astronomy.mps.ohio-state.edu/~pogge/Ast162/Movies/specbin.html

Like with planets; can get M sin(i), but dont know i

(c) AESOhttp://esomac.as.utexas.edu:81/database/slides/TB/F00001.html
http://www-astronomy.mps.ohio-state.edu/~pogge/Ast162/Movies/specbin.htmlhttp://www-astronomy.mps.ohio-state.edu/~pogge/Ast162/Movies/specbin.htmlhttp://www-astronomy.mps.ohio-state.edu/~pogge/Ast162/Movies/specbin.htmlhttp://www-astronomy.mps.ohio-state.edu/~pogge/Ast162/Movies/specbin.html
Visual Eclipsing Spectroscopic Binaries Know everything,

because i ~ 90 degrees

Know total mass, mass ratio (why?)

Can get separation

If can see separation, know distance

P 3 =42

G(m1 + m2)a3

D = a()1

D

a

Pulsating Stars

Photometric distance where the intrinsic luminosity is given to you indirectly

Luminosity drives radial pulsations

Galactic Dynamics

CO Fig 24.19

Starting point; average conditions in Suns neighborhood

Sun might be drifting w/rt local motions

Local Standard of Rest (LSR): perfectly circular, non-vertical orbit at Suns position.

LSR

Should be easy to find Suns motion w/rt LSR

Average motions should be LSR (?)

Just find Suns relative velocity to average motions in the neighborhood.

Velocity w/rt neighborhood =

LSRBut this doesnt work

Average motions dont cancel out

Stars passing through from inner orbits moving faster

But then correlation between radial velocity and angular offset...

CO Fig 24.30

LSR still works for u (radial), w (vertical) velocities

for v, must make correction

Has to be statistical; dont know where star is in its ellipsoidal phase

But whats C ? - can measure

CO Fig 24.30

< v >= C < u2 >"= 0

Why ~?

Stars born on nearly circular orbit

Interactions with (stars, gas, spiral arms) `scatter slightly

Kinetic energy tends to be roughly conserved

Why ~?

Off by a factor of 2, but right idea;More details in section on spiral arms

KE =12M!

(u2 + (v + vc)2 + w2

)

u2 + (v + vc)2 + w2 0(v + vc)2 2u2

(v2c + 2vcv + v

2) 2u2

2vcv + v2 2u2

v u2

vc

LSR

Once this is done, can measure Suns velocity w/rt LSR

(u,v,w) ~ (-10,5,7) km/s Moving inwards, upwards, and slightly faster

in direction of rotation than LSR

LSR Yes, but we still dont know the LSRs motion. Can infer orbital velocity

from halo stars

In LSR, huge distribution of old stars (Pop II) in velocity space around LSR

Goes much larger to -ve velocities

Centred around -220 km/s

220 km/s must be local disk velocity

Gives us a distance to the galactic centre!

Standard values; 220 km/s, 8.5 kpc.

Joshua Barnes,http://www.ifa.hawaii.edu/~barnes/ast110_01/

CO, Fig 24.21
Less Local Now have LSR down

But want to understand the whole galaxy!

First understand the larger neighborhood -- how does the SR vary with radius?

Assume everything on circular orbits (on average)

R0

R

v(R0) = 220 km/s

l

v(R) = ?

D

`Solid Body


Flat Rotation curve

Keplerian

vr r1/2

r3/2

= constvr = r r

But for anyreasonablerotation curve,angularvelocity

goes up as radius gets

smaller

CO Fig 24.23Vr, Vt as a function of l

The different amplitudesand offsets here are going to

tell us about our local rotation curve.

Less Local

R0

R

v(R0) = 220 km/s

l

v(R) = ?

D

vr = v(R) cos v(R0) sin lvt = v(R) sin v(R0) cos l

Radial velocity difference is greatest for objects moving along line of sight -- youre catching it at the tangent point of their orbits.

Objects Motion

Suns Motion

Tangent Point Method

CO Fig 24.24

But can only use local stars

R0

R

v(R0) = 220 km/s

l

v(R) = ?

D

Measuring Rotation Curve

of disk means looking in disk

Extinction

Can correct for it over short distances, but cant see anything beyond those

But dust largely transparent to lines from atomic gas (eg, 21 cm)


Careful measurements of molecular line emission in plane gives max velocity for each angle

For each angle, know R;

Know rotation curve! 12CO lines from molecular gas cloudsEnglmaier & Gerhard (1999) MNRAS 304:512

sin l = R/R0

Weighing our Galaxy

From rotation curve at LSR, can find mass interior

88 Billion Solar Masses

Rotation Period ~230 Myr

From largely-flat rotation curve can figure out density profile beyond LSR.

vr(R) =

GM(< R)

r(220 km s1

)2 = GM(< R)8.5 kpc

M(< R) =8.5 kpc

(220 km s1

)2

G= 8.8 1010M"

Rotation curve outside of the Solar Circle

Cant use tangent point method for larger R

for

Need to keep original form that involves D and use objects can measure D to directly

vr = ( 0)R0 sin l

0 > R > R0

Harvard/CfA scientists examined masers (microwave lasers) which occur in star forming regions

Bright (all radiation is in one line), not extinguished by dust

Can get velocites; distances by parallax (but how?)

New Results

Monday

More on Rotation curves: Accurately characterizing the rotation curve locally -- Oort

Constants

CO 909-914

Rotasi Galaksi
Piringan galaksi tidak berotasi seperti benda tegarBintang yg lebih dekat ke pusat galaksi berotasi dg lebih cepatw(R) tidak konstanDikenal dg rotasi diferensialKecepatan rotasi :Exponential disk (full line)Spherical (dashed)Point mass (dotted)Untuk distribusi massa spherical
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Rotasi diferensial bintang di galaksi

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Kecepatan bintang relatif thd LSR

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Vektor kecepatan radial

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Chart10102030405060708090100110120130140150160170180190200210220230240250260270280290300310320330340350360Bujur galaksi l [deg]Vr [km/s]efek rotasi diferensial thd kec. radial05.472322293210.28460175513.856406460615.756924048215.756924048213.856406460610.2846017555.47232229320-5.4723222932-10.284601755-13.8564064606-15.7569240482-15.7569240482-13.8564064606-10.284601755-5.4723222932-05.472322293210.28460175513.856406460615.756924048215.756924048213.856406460610.2846017555.47232229320-5.4723222932-10.284601755-13.8564064606-15.7569240482-15.7569240482-13.8564064606-10.284601755-5.4723222932-0Sheet100105.47232229322010.2846017553013.85640646064015.75692404825015.75692404826013.85640646067010.284601755805.4723222932900100-5.4723222932110-10.284601755120-13.8564064606130-15.7569240482140-15.7569240482150-13.8564064606160-10.284601755170-5.4723222932180-01905.472322293220010.28460175521013.856406460622015.756924048223015.756924048224013.856406460625010.2846017552605.47232229322700280-5.4723222932290-10.284601755300-13.8564064606310-15.7569240482320-15.7569240482330-13.8564064606340-10.284601755350-5.4723222932360-0Sheet10000000000000000000000000000000000000l [deg]Vr [km/s]efek rotasi diferensial thd kec. radial0000000000000000000000000000000000000Sheet2Sheet3

Vektor kecepatan tangential

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Chart10102030405060708090100110120130140150160170180190200210220230240250260270280290300310320330340350360Bujur galaksi l [deg]Vtan [km/s]efek rotasi diferensial thd kec. tangential76.03508193263.2567110899-1-6.2216291573-11.7783708427-17-21.2567110899-24.0350819326-25-24.0350819326-21.2567110899-17-11.7783708427-6.2216291573-13.25671108996.035081932676.03508193263.2567110899-1-6.2216291573-11.7783708427-17-21.2567110899-24.0350819326-25-24.0350819326-21.2567110899-17-11.7783708427-6.2216291573-13.25671108996.03508193267Sheet1007105.47232229326.03508193262010.2846017553.25671108993013.8564064606-14015.7569240482-6.22162915735015.7569240482-11.77837084276013.8564064606-177010.284601755-21.2567110899805.4723222932-24.0350819326900-25100-5.4723222932-24.0350819326110-10.284601755-21.2567110899120-13.8564064606-17130-15.7569240482-11.7783708427140-15.7569240482-6.2216291573150-13.8564064606-1160-10.2846017553.2567110899170-5.47232229326.0350819326180-071905.47232229326.035081932620010.2846017553.256711089921013.8564064606-122015.7569240482-6.221629157323015.7569240482-11.778370842724013.8564064606-1725010.284601755-21.25671108992605.4723222932-24.03508193262700-25280-5.4723222932-24.0350819326290-10.284601755-21.2567110899300-13.8564064606-17310-15.7569240482-11.7783708427320-15.7569240482-6.2216291573330-13.8564064606-1340-10.2846017553.2567110899350-5.47232229326.0350819326360-07Sheet10000000000000000000000000000000000000l [deg]Vr [km/s]efek rotasi diferensial thd kec. radial0000000000000000000000000000000000000Sheet20000000000000000000000000000000000000l [deg]Vtan [km/s]efek rotasi diferensial thd kec. tangential0000000000000000000000000000000000000Sheet3

GC

Sun

Star

Ro

R

r

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Konstanta Oort

Data kecepatan radial bintang-bintang kelas spektrum A

untuk menghitung konstanta Oort A

(setelah dikoreksi oleh jarak-rata-rata)

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Konstanta Oort

Distribusi bintang cepheid dan

data gerak dirinya, untuk

Menghitung konstanta Oort A dan B

Distribusi ruang

Gerak diri

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Kecepatan rotasi di sekitar Matahari

wo=Vc/Ro=A-B

Rotasi diferensial di sekitar Matahari

A+B=dVc/dR|Ro

Solar Motion :

dengan arah Apex :

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49.pdf
Wednesday Extragalactic astronomy tools

Multi-wavelength astronomy Starcounts Extinction Globular Clusters

Milky way components: Disk: exponential profile, thin/thick disk Stellar Halo, Pop I/II stars, Globular Clusters Age-Metallicity Relation ISM - HII (Hot Ionized), HI (atomic), H2 (molecular) Bulge Dark Matter Halo Satellite galaxies

Kinematics of the Galaxy I - Rotation

CurvesJL 1.3,1.6; Carroll & Ostlie pp 898-907, 915-933

Today Disk, Halo Motions Galactic Coordinates Measuring Distance: parallax pulsating stars Rotation Curves Local Standard of Rest Rotation speed from halo star motions Tangent-Point Method and Rotation Curves Weighing in New results?

Chung et al:http://www.astro.columbia.edu/~archung/research/CRdisk/crdisk.html

Other Galaxies Motions

Radial velocities can be measured with spectra

Can get rotation curves, (radial) velocity dispersions
http://www.astro.columbia.edu/~archung/research/CRdisk/crdisk.htmlhttp://www.astro.columbia.edu/~archung/research/CRdisk/crdisk.html
Where are we and how fast

are we moving? Within our own galaxy, more

difficult! Even though we know were

edge on, even local position and rotation speed are uncertain

The Open University

30 kpc

6 kpc

1

50 kpc


bulge

(a)

(b)

stellar disc

bulge

stellardisc

locationof Sun

locationof Sun

1 kpc

gaseousdisc


JL Fig 1.5

Even now, numbers

jumping around Recent (announced Tues at

AAS meeting) found evidence of ~15% higher rotation rate than generally understood

Using parallax of very bright microwave objects (masers)

Rotation curves of other spirals

Rotation curves of spiral galaxies quite generally have flat rotation curves

Non-visible matter contributing to mass of galaxies, not to light

Dark Matter

Sofue et al (1997)http://www.ioa.s.u-tokyo.ac.jp/~sofue/rotation/fig1/all-new.gif
http://www.ioa.s.u-tokyo.ac.jp/~sofue/rotation/fig1/all-new.gifhttp://www.ioa.s.u-tokyo.ac.jp/~sofue/rotation/fig1/all-new.gifhttp://www.ioa.s.u-tokyo.ac.jp/~sofue/rotation/fig1/all-new.gifhttp://www.ioa.s.u-tokyo.ac.jp/~sofue/rotation/fig1/all-new.gif
M

v

r

r

vr

8

Circular velocity around

a mass Assuming symmetry, constant

circular orbit around a (spherical) mass distribution only depends on total mass enclosed:

v2r(R) =GM(< R)

r

`Solid Body


Flat Rotation curve

Keplerian

vr r1/2

r3/2

Examples of

differential rotation

= constvr = r r


Sgr A* shows very regular proper motion, even though surrounded by thousands of starsMotions entirely consistent with being due only to solar motion


v! = (0.4 0.9)km s1

But dont know distance from galactic centre; cant automatically turn this into a rotational velocity

RA & Dec for Solar System

ESO: http://www.eso.org/public/outreach/eduoff/vt-2004/Background/Infol2/EIS-D3.html
http://www.eso.org/public/outreach/eduoff/vt-2004/Background/Infol2/EIS-D3.htmlhttp://www.eso.org/public/outreach/eduoff/vt-2004/Background/Infol2/EIS-D3.html

CO Fig 24.17


CO Fig 24.19

VU

Galactic Orbits

Stars in disk typically have aligned circular orbits

In halo, orbits fill halo

PopI vs PopIIJoshua Barnes,

http://www.ifa.hawaii.edu/~barnes/ast110_01/
Parallax

1 AUp

D/tan(p) ~ D/p

Definition: 1pc = (1 AU)/(1 (in radians))

But measuring them is hard

First parallax measurement: Bessel (1939)

61 Cygni

Measured parallax of 0.31360.015 (d = 3.04-3.35 pc)

Current distance: 3.496 0.007 pc

Hipparcos mission

For stars larger than few parsec away, limits on seeing prevent good measurements

118,000 stars with high-precision distances, proper motions

Still limited by baseline - ~1 AU Hipparcos: 1989-1993

http://www.rssd.esa.int/index.php?project=HIPPARCOS
http://www.rssd.esa.int/index.php?project=HIPPARCOShttp://www.rssd.esa.int/index.php?project=HIPPARCOS
Photometric Distances

Once know stars distances in local neighborhood, know real luminosity (L) and absolute magnitude M

From observed m, can get distance modulus and distance

But extinction! m is less than it would be with just distance effect.

d = (10pc)10(mM)/5


Depends on wavelength

So by finding out how much the difference is in different wavelengths, can estimate the magnitude of effect

So can still get a distance

Alves, Lada & Lada (2001)

Visible

Near IR

d = (10pc)10(mMA)/5


Sketch of how extinction behaves

Less effective at low frequencies

Some similarities to scattering of light in atmosphere

(eg, why is sky blue?)http://www.jb.man.ac.uk/distance/life/sample/stars/index.html

Jodrell Bank Observatory
http://www.jb.man.ac.uk/distance/life/sample/stars/index.htmlhttp://www.jb.man.ac.uk/distance/life/sample/stars/index.html
Spectroscopic Binaries

R. Poggehttp://www-astronomy.mps.ohio-state.edu/~pogge/Ast162/Movies/specbin.html

Like with planets; can get M sin(i), but dont know i

(c) AESOhttp://esomac.as.utexas.edu:81/database/slides/TB/F00001.html
http://www-astronomy.mps.ohio-state.edu/~pogge/Ast162/Movies/specbin.htmlhttp://www-astronomy.mps.ohio-state.edu/~pogge/Ast162/Movies/specbin.htmlhttp://www-astronomy.mps.ohio-state.edu/~pogge/Ast162/Movies/specbin.htmlhttp://www-astronomy.mps.ohio-state.edu/~pogge/Ast162/Movies/specbin.html
Visual Eclipsing Spectroscopic Binaries Know everything,

because i ~ 90 degrees

Know total mass, mass ratio (why?)

Can get separation

If can see separation, know distance

P 3 =42

G(m1 + m2)a3

D = a()1

D

a

Pulsating Stars

Photometric distance where the intrinsic luminosity is given to you indirectly

Luminosity drives radial pulsations

Galactic Dynamics

CO Fig 24.19

Starting point; average conditions in Suns neighborhood

Sun might be drifting w/rt local motions

Local Standard of Rest (LSR): perfectly circular, non-vertical orbit at Suns position.

LSR

Should be easy to find Suns motion w/rt LSR

Average motions should be LSR (?)

Just find Suns relative velocity to average motions in the neighborhood.

Velocity w/rt neighborhood =

LSRBut this doesnt work

Average motions dont cancel out

Stars passing through from inner orbits moving faster

But then correlation between radial velocity and angular offset...

CO Fig 24.30

LSR still works for u (radial), w (vertical) velocities

for v, must make correction

Has to be statistical; dont know where star is in its ellipsoidal phase

But whats C ? - can measure

CO Fig 24.30

< v >= C < u2 >"= 0

Why ~?

Stars born on nearly circular orbit

Interactions with (stars, gas, spiral arms) `scatter slightly

Kinetic energy tends to be roughly conserved

Why ~?

Off by a factor of 2, but right idea;More details in section on spiral arms

KE =12M!

(u2 + (v + vc)2 + w2

)

u2 + (v + vc)2 + w2 0(v + vc)2 2u2

(v2c + 2vcv + v

2) 2u2

2vcv + v2 2u2

v u2

vc

LSR

Once this is done, can measure Suns velocity w/rt LSR

(u,v,w) ~ (-10,5,7) km/s Moving inwards, upwards, and slightly faster

in direction of rotation than LSR

LSR Yes, but we still dont know the LSRs motion. Can infer orbital velocity

from halo stars

In LSR, huge distribution of old stars (Pop II) in velocity space around LSR

Goes much larger to -ve velocities

Centred around -220 km/s

220 km/s must be local disk velocity

Gives us a distance to the galactic centre!

Standard values; 220 km/s, 8.5 kpc.

Joshua Barnes,http://www.ifa.hawaii.edu/~barnes/ast110_01/

CO, Fig 24.21
Less Local Now have LSR down

But want to understand the whole galaxy!

First understand the larger neighborhood -- how does the SR vary with radius?

Assume everything on circular orbits (on average)

R0

R

v(R0) = 220 km/s

l

v(R) = ?

D

`Solid Body


Flat Rotation curve

Keplerian

vr r1/2

r3/2

= constvr = r r

But for anyreasonablerotation curve,angularvelocity

goes up as radius gets

smaller

CO Fig 24.23Vr, Vt as a function of l

The different amplitudesand offsets here are going to

tell us about our local rotation curve.

Less Local

R0

R

v(R0) = 220 km/s

l

v(R) = ?

D

vr = v(R) cos v(R0) sin lvt = v(R) sin v(R0) cos l

Radial velocity difference is greatest for objects moving along line of sight -- youre catching it at the tangent point of their orbits.

Objects Motion

Suns Motion


CO Fig 24.24

But can only use local stars

R0

R

v(R0) = 220 km/s

l

v(R) = ?

D

Measuring Rotation Curve

of disk means looking in disk

Extinction

Can correct for it over short distances, but cant see anything beyond those

But dust largely transparent to lines from atomic gas (eg, 21 cm)


Careful measurements of molecular line emission in plane gives max velocity for each angle

For each angle, know R;

Know rotation curve! 12CO lines from molecular gas cloudsEnglmaier & Gerhard (1999) MNRAS 304:512

sin l = R/R0

Weighing our Galaxy

From rotation curve at LSR, can find mass interior

88 Billion Solar Masses

Rotation Period ~230 Myr

From largely-flat rotation curve can figure out density profile beyond LSR.

vr(R) =

GM(< R)

r(220 km s1

)2 = GM(< R)8.5 kpc

M(< R) =8.5 kpc

(220 km s1

)2

G= 8.8 1010M"

Rotation curve outside of the Solar Circle

Cant use tangent point method for larger R

for

Need to keep original form that involves D and use objects can measure D to directly

vr = ( 0)R0 sin l

0 > R > R0

Harvard/CfA scientists examined masers (microwave lasers) which occur in star forming regions

Bright (all radiation is in one line), not extinguished by dust

Can get velocites; distances by parallax (but how?)

New Results

Monday

More on Rotation curves: Accurately characterizing the rotation curve locally -- Oort

Constants

CO 909-914

CO

HI

*

To map out vr throughout Galaxy, divide the Galaxy into quadrants based on value of galactic longitude.

Quad I (ll>90) - all los pass through orbits outside of the Suns. No maximum vr but increases with d.

Quad III (270>l>180) - similar to Quad II but opposite signs.

Quad IV (l>270) - similar to Quad I except reverse signs.

*

Orbital Radius

*

Vc2 = Vc,b2 + Vc,d2 + Vc,g2 + Vc,h2

*

Bagaimana menghitung massa Galaksi ?
Dari kurva rotasi kita dapat menghitung massa Galaksi MGal= 8,8 x 1010 MOPeriode rotasi = 230 x 106 tahun
*

R

)

R

(

GM

V

c

=

2

o

90

=

l

o

0

=

l

o

90

=

l

efek rotasi diferensial thd kec. radial

090180270360

Bujur galaksi l [deg]

Vr [km/s]

efek rotasi diferensial thd kec.

tangential

090180270360

Bujur galaksi l [deg]

Vtan [km/s]

l

cos

R

r

cos

R

l

sin

R

sin

R

l

cos

V

cos

V

V

l

sin

V

sin

V

V

t

r

0

0

0

0

=

+

a

=

a

-

a

=

-

a

=

(

)

(

)

r

l

cos

R

V

l

sin

R

V

t

r

w

-

w

-

w

=

w

-

w

=

0

0

0

0

l

a

=

R

V

w

R

V

w

=

rotasi galaksi 2011

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