Radial migration of the Sun in galactic disk

0.19 MB·English

by  J. Klacka

Radial migration of the Sun in galactic disk
J. Klaˇcka, M. Jurˇci, J. Durˇzo and R. Nagy
Faculty of Mathematics, Physics, and Informatics, Comenius University,Mlynsk´a dolina, 842 48 Bratislava, Slovakia
e-mails: [email protected] ; [email protected] ; [email protected] ; [email protected]
2
1
ABSTRACT
0
2
Physics of the gravitational effect of the galactic bar and spiral structure is pre-

n sented. Physical equations differ from the conventionally used equations.
a Application to the motion of the Sun is treated. The speed of the Sun is taken to
J
be consistent with the Oort constants.

1 Galactic radial migration of the Sun is less than ± 0.4 kpc for the four-armed

spiral structure. The Sun remains about 75 % of its existence within galactocentric
A] distances (7.8 − 8.2) kpc and the results are practically independent on the spiral
structure strength. Thus, the radial distance changes only within 5% from the value
G
of 8 kpc.
h. Galactic radial migration of the Sun is less than ± (0.3 − 1.2) kpc, for the two-
p armedspiralstructure.TheSunremains(29−95)%ofitsexistencewithingalactocen-
- tric distances (7.8 − 8.2) kpc and the results strongly depend on the spiral structure
o
strengthandthe angularspeed ofthe spiralarms.The radialdistance changeswithin
tr (3.8 − 15.0)% from the value of 8 kpc.
s
If observational arguments prefer relevant radial migration of the Sun, then the
a
[ Milky Way is characterized by the two-arm spiral structure.

1 Key words: Galaxy – galactic bar – galactic spirals – motion of the Sun – equation
v of motion.
0
7
3
0
1 INTRODUCTION potential and the concentration is paid to the quadrupole
.
1 term,importantforthegalacticbar.Therelevantquantities
0 OrbitalevolutionoftheSuninourgalaxy,MilkyWay,isim- found in the monopole and quadrupole terms are discussed
2 portant for understanding of galactic evolution, its dynam-
inSection3,wherethevaluesofthequantitiesarecalculated
1 ics,kinematicsandchemicalcomposition.Itisalsorelevant
for several models of the galactic bar. Section 4 offers the
: for understandingof evolution of theSolar System,e.g. the
v equation of motion for an object under the gravitational
effect of the galactic tides on the evolution of the O¨pik -
i influence of the galactic disk, halo and the monopole and
X Oort cloud of comets. Several papers on the gravitational
quadrupole terms of the galactic bar (the monopole term
r effectofthegalacticbaronmotionoftheSunintheGalaxy
corresponds to the conventional idea of the galactic bulge).
a appeared recently, e.g. Minchev & Famaey (2010), Famaey
Section5presentsnumericalresultsfortheradialmigration
& Minchev (2010), Minchev et al. (2010), Minchev et al.
of the Sun. Finally, discussion and conclusion can be found
(2011). One of the important results of the papers shows
in Sections 6 and 7.
thattheSuncan radially migrateinthediskoftheGalaxy.
Minchev & Famaey (2010), Famaey & Minchev (2010)
present thechangeofthesolar galactocentric distance.The
distance may change within several kiloparsecs.
If we are interested in real galactic radial migration of 2 MULTIPOLE EXPANSION
the Sun,we haveto treat physical equation of motion. The
Having a mass density distribution ̺(~r), the gravitational
first aim of this paper is to find the equation of motion, to
potential at a point~r is
puttherelevantequationsonafirmphysicalbasis.Thesec-
ondaimofthispaperistousethephysicalequationsinper- ̺(~r ′)
forming calculations similar to Minchev & Famaey (2010). Φ(~r)= G d3~r ′ , (1)
− ~r ~r ′
Weare interestedin radialmotion (radial migration) ofthe Z | − |
Sun in the Galaxy. whereGisthegravitationalconstant.Makinganexpansion
Section 2 presents multipole expansion of the galactic in terms of theratio ~r ′/~r r′/r, we can write
| | | | ≡
2 J. Klaˇcka et al.
∞
Inserting Eqs. (9) into Eq. (7) oneobtains
Φ(~r)= Φ(j)(~r) , (2)
G Q (1 Q /Q )
Xj=0 Φ(2)(x,y,z) = − 11 4−r5 22 11 × XΦ2
and,
1+Q /Q
X = 22 11 x2+y2 2 z2
Φ(0)(~r)= G 1 ̺(~r ′) d3~r ′= G M , (3) Φ2 1−Q22/Q11 −
− r Z − r + x2 y2 c(cid:0)os(2α) + 2 x(cid:1)y sin(2α) ,
−
whereM isthetotalmassconcentratedinasmallregionso α = Ω t + α , (10)
that the condition ~r ′/~r 1 is fulfilled. Other terms of b(cid:0) 0(cid:1)
the expansion are Φ|(1)(|~r|)|=≪0, and, the quadrupole term is ifalsoEq.(6)isadded.AsfortheMilkyWay,thevalueofα0
(Einstein’s summation convention is used) is about 25 degrees (compare with Majaess 2010; the real
−
value may differ in more than 10 degrees) and the value of
Φ(2)(~r) = − 2Gr3 Qij ni nj , theangularvelocityisΩb =−55.5kms−1 kpc−1 (compare
with Minchev & Famaey 2010), t is the time.
Q = [3(~r ′) (~r ′) ~r ′2δ ] ̺(~r ′) d3~r ′ ,
ij i j ij
− | |
Z
n = (~r) /r (4) 2.1 Discussion on Eq. (10)
i i
and δ is the Kronecker delta. Since Φ(2)(~r) is given by a Our result represented by Eq. (10) differs from the result
ij
contractionoftwotensorsofthesecondorder,Q andn n , conventionally used by other authors. The conventional re-
ij i j
it is an invariant independent on the coordinate basis. The sult is (see, e.g., Minchev & Famaey 2010, Dehnen2000)
numerical calculations can be done in a special coordinate
1
basis, the primed coordinates, e.g., defined by the relations Φ(c2o)nv = 2 QT vc2 cos[2(φ−Ωbt)]×XΦ2c ,
describing rotation
r 3
X = b , r>r ,
x′ = + x cosα + y sinα , Φ2c r b
y′ = x sinα + y cosα , X = 2(cid:16) (cid:17)r 3 , r6r , (11)
z′ = z−. (5) Φ2c − rb b
(cid:16) (cid:17)
where r = 3.44 kpc, v = 240 km s−1, 0.1 < Q < 0.4
In the case of a galactic bar we will use b c T
(Minchev & Famaey 2010).
α=Ω t + α , (6) At first,thesecond formula of Eqs. (11)does not hold.
b 0
The inner part of the galactic bar potential must be calcu-
where Ω is the magnitude of the angular velocity of the
| b| lated from the Poisson equation or its equivalence: Φ =
bar’s rotation. In reality, Ωb is negative and the negative 4πG̺, Φ = G ̺(~r ′)/~r ~r ′ d3~r ′. As for th△e case
signdenotestheobservationalfactthatrotationisnegative, − | − |
r>r in Eqs. (11),it leads to
i.e., clockwise. b R
Eqs. (4)-(6) yield 1 r 3 1
Φ(2) (x,y,z) = Q v2 b Y
Φ(2)(x,y,z) = − 2Gr3 × XΦ , conv Y = 2x2 T y2c (cid:18)c√osR(22Ω+tz)2(cid:19) R2 × Φ2c
Φ2c b
X = Q n2+Q n2 (Q +Q )n2 , (7) −
Φ 11 1 22 2− 11 22 3 (cid:0)+ 2 x y (cid:1)sin(2Ωbt) . (12)
whereprimesabovetheQ andQ termsareomitted,for
11 22 Weremindthatr=√R2+z2,whereRisthe2-dimensional
the purpose of brevity. The coordinate system is chosen in
radial coordinate in the galactic equatorial plane and z is
thewaythatQ =0ifi=jandsomesymmetryexists.The
ij 6 thevertical coordinate normal to theequatorial plane. One
mass density is an even function of coordinate arguments,
can immediately see that physical Eq. (10) differs from the
∞
conventional potential.
Q = ρ(x,y,z) 2x2 y2 z2 dxdydz ,
11
− −
Z
−∞ (cid:0) (cid:1)
∞ 3 MODELS OF THE GALACTIC BAR
Q = ρ(x,y,z) 2y2 x2 z2 dxdydz (8)
22 − − Asexamplesofmassdistributioninthegalacticbarwetake
Z
−∞ (cid:0) (cid:1) themodels presented by Staneket al. (1997). For themod-
and elsG1, G3,E2, P1,P2, P3weintroducethefollowing com-
plete nonsymmetric ellipsoid coordinates (r is a dimension-
n2 = x2cos2α+y2sin2α+2xysinαcosα /r2 ,
1 less quantity,now – it differs from the quantity used in the
n22 = (cid:0)x2sin2α+y2cos2α−2xysinαcosα(cid:1)/r2 , previous section):
n23 = (cid:0)z2/r2 . (cid:1) (9) x = r x0 sinθ cosϕ ,
Radial migration of the Sun in galactic disk 3
y = r y sinθ sinϕ , yields
0
z = r z0 cosθ , (13) rbar
r2
with thecorresponding volume element M = 4π ρ0 x0 y0 z0 (1+r)4 dr
Z
dx dy dz=x y z r2 sinθ dθ dϕdr. (14) 0
0 0 0
4π r3
The corresponding quantities Q and Q , and, also mass = ρ x y z bar ,
11 22 3 0 0 0 0 (1+r )3
ofthegalacticbarM,cannowbecalculatedinananalytical bar
way. rbar
4π r4
Q = ρ x y z 2x2 y2 z2 dr
(i) The model G1 with mass density 11 3 0 0 0 0 0− 0− 0 (1+r)4
Z
(cid:0) (cid:1) 0
ρG1(x,y,z)=ρ0exp − r2/2 , = 4π ρ x y z 2x2 y2 z2 X
3 0 0 0 0 0− 0− 0 × P1
r= x 2+ y(cid:0) 2+ (cid:1)z 2 (15) = 1 M 2x2 y2(cid:0) z2 3r +(cid:1)22+ 30
yields s(cid:16)x0(cid:17) (cid:18)y0(cid:19) (cid:16)z0(cid:17) 3 12(cid:0) 0− 0− 0(cid:1)h1 ba3r rbar
+ 12 1+ ln(1+r ) ,
r2 − r bar
M = 2 √2 π3/2 ρ0 x0 y0 z0 , bar (cid:16) bar(cid:17) (cid:21)
Q11 = 2 √2 π3/2 ρ0 x0 y0 z0 2x20−y02−z02 XP1 = 3rb4ar+223rb3(a1r++r30rb)2a3r+12rbar −4ln(1+rbar)
= M 2x20−y02−z02 , (cid:0) (cid:1) r r /x (as an exabamrple). (22)
Q 2y2 x2 z2 bar ≡ b 0
22 = 0(cid:0)− 0− 0 . (cid:1) (16)
Q11 2x20−y02−z02 iWnteeghraavleintaEkqe.n(2th2e) (siyfmonbeolwrobuarldalsiktehteouupspeeirnfilinmitiyt oafs tthhee
(ii) The model G3 defined bythe mass density
upperlimit of theintegral, then theintegral would diverge,
ρG3(x,y,z)=ρ˜0 r−1.8exp − r3 , (17) Qba1r1r→i∞sp.rIofpwoerttiaoknealinttooxac,ctohuenntwtheactatnhetarkeaelrleng=throf/txhe.
b 0 bar b 0
produces (cid:0) (cid:1) (v) The model P2 with mass density
M =. 44ππ×0.7394×ρ˜0 x0 y0 z0 , ρP2(x,y,z)= r(1ρ+0r)3 (23)
Q = 0.3219 ρ˜ x y z 2x2 y2 z2
11 3 × × 0 0 0 0 0− 0− 0
yields
.
= 0.1451 M 2x2 y2 z2(cid:0) , (cid:1)
× 0− 0− 0 rbar
QQ22 = 22xy022−xy220−zz0(cid:0)22 , (cid:1) (18) M = 4π ρ0 x0 y0 z0 (1+rr)3 dr
11 0− 0− 0 Z
0
where the number 0.3219 is an approximate value of the
integral = 2π ρ x y z rb2ar ,
0 0 0 0 (1+r )2
∞ bar
r2.2exp r3 dr. 4π rbar r3
− Q = ρ x y z 2x2 y2 z2 dr
Z0 (cid:0) (cid:1) 11 3 0 0 0 0 0− 0− 0 Z (1+r)3
(cid:0) (cid:1) 0
(iii) The model E2 definedby themass density
4π
= ρ x y z 2x2 y2 z2 X
ρ (x,y,z)=ρ exp( r) (19) 3 0 0 0 0 0− 0− 0 × P2
E2 0
− = 1 M 2x2 y2(cid:0) z2 2r +(cid:1)9+ 6
gives 3 0− 0− 0 bar r
bar
M = 8π ρ x y z , (cid:0) 1 2 (cid:1)h
0 0 0 0 6 1+ ln(1+r ) ,
Q11 =. 32π ρ0 x0 y0 z0 2x20−y02−z02 − (cid:16) rbar(cid:17) bar (cid:21)
Q22/Q11 == 422xyM0220−−(cid:0)xy20x22020−−−zz0022y02.−(cid:0) z02(cid:1) , (cid:1) (20) XrbPa2r =≡ r2br/b3xar02+(1(9+arsb2raabran+r)e26xrabmarp−le)3.ln(1+rbar) (24)
(iv) The model P1 with mass density (vi) The model P3 with mass density
ρ ρ
ρ (x,y,z)= 0 (21) ρ (x,y,z)= 0 (25)
P1 (1+r)4 P3 (1+r2)2
4 J. Klaˇcka et al.
yields for the lower hemisphere, and, for both hemispheres r
s
∈
0, ),ϑ 0,π/2 ,ϕ 0,2π),and,thevolumeelementis
rbar h ∞ ∈h i ∈h
r2
M = 4π ρ0 x0 y0 z0 Z (1+r2)2 dr dx dy dz= 21 x0 y0 z0 rs2 cos−1/2θ dθ dϕdrs (34)
0
for both hemispheres. The values of the quantities M, Q
r 11
= 2π ρ0 x0 y0 z0 arctanrbar− 1+barr2 , and Q22/Q11 are:
(cid:18) bar(cid:19)
Q = 4π ρ x y z 2x2 y2 z2 rbar r4 dr M = 2 π3/2 K 12 ρ0 x0 y0 z0 ,
11 3 0 0 0 0 0− 0− 0 (1+r2)2 Q = 6π3/2√2 ρ(cid:16) x(cid:17) y z X
Z 11 0 0 0 0 G2
= 43π ρ0 x0 y0 z(cid:0)0 2x20−y02−z(cid:1)02 0 × = 3√2 K 12 −1M ××XG2,
r + 1 rba(cid:0)r 3arctan(cid:1)r Q22 = 2y02−hx20(cid:16)−(cid:17)i2/π[Γ(3/4)]2z02 ,
(cid:18) bar 21+rb2ar − 2 bar(cid:19) Q11 2x20−y02−p2/π[Γ(3/4)]2z02
2
= M 2x2 y2 z2 p2 3 2
3 0− 0− 0 × XG2 = 2x20−y02− π Γ 4 z02 , (35)
rbar+(cid:0)rbaarrc/ta2nr1+r(cid:1)b2arr /−(1(3+/2r)2ar)ctanrbar , whereK(x)isthecomrpletehelli(cid:16)pti(cid:17)ciintegralofthefirstkind.
(cid:2) (cid:0)bar− ba(cid:1)r(cid:3) bar Finally, themodel E3
r r /x (as anexample). (26)
bar ≡ b 0 ρE3(x,y,z)=ρ0 K0(rs) , (36)
Asfor themodel E1 with mass density
yields
ρ (x,y,z)=ρ exp( r ), (27)
E1 0 e 1
− M = √2 π2 K ρ x y z ,
where 2 0 0 0 0
x y z Q11 = 18 π2 ρ0 x(cid:16)0 y(cid:17)0 z0 XE3
re = |x0| + |y0| + |z0|, (28) = 9√2 K 1 −1M× X ,
2 × E3
thevaluesofM,Q aQ can beeasily analytically calcu-
lated (one may avo1i1d Eqs2.213 – 14): Q22 = 2y02−hx20(cid:16)−(cid:17)i2/π[Γ(3/4)]2z02 ,
M = 8 ρ0 x0 y0 z0 , Q11 2x20−y02−p2/π[Γ(3/4)]2z02
Q11 == 126Mρ0x20xy200z−0y(cid:0)022x−20z−02y02,−z02(cid:1) XE3 = 2x20−y02−rpπ2 hΓ(cid:16)34(cid:17)i2z02 . (37)
2y2 x2 z2
Q22/Q11 = 2x02−(cid:0)y20−z02 . (cid:1) (29)
0− 0− 0
3.1 Numerical results
The model G2 with mass density
Table 1 presents numerical results for the data given by
ρ (x,y,z)=ρ exp r2/2 , (30)
G2 0 − s Stanek et al. (1997 – Table 4). Since the integrations in
where (cid:0) (cid:1) the models P1, P2 and P3 lead to integrals which require
2 2 1/4 finite limits in order to be convergent, we have considered
x 2 y z 4 the upper limit corresponding to the length of the bar r
r = + + , (31) b
s ("(cid:16)x0(cid:17) (cid:18)y0(cid:19) # (cid:16)z0(cid:17) ) 2a6n)d.tThaekvinaglueinstoofaxc0cofourntintdhiveidvualaulemrod=els3(.s4e4ekEpqcs,.t2h2e, 2r4e-,
b
can be solved with the following substitutions: quired quantities are presented in Table 2. If we admit a
x = x r sin1/2ϑ cosϕ , relativeerrorofrbtobe20%,thentherelativeerrorsofthe
0 s
quantities are also presented in Table 2.
y = y r sin1/2ϑ sinϕ ,
0 s
z = z r cos1/2ϑ , (32)
0 s
for theupperhemisphere, and,
4 CONVENTIONAL AND PHYSICAL
x = x r sin1/2ϑ cosϕ , APPROACHES
0 s
y = y0 rs sin1/2ϑ sinϕ , Inordertocomparethepublishedresultswithourapproach,
z = z r cos1/2ϑ , (33) we haveto summarize important equations.
0 s
−
Radial migration of the Sun in galactic disk 5
Table1.Numericalvaluesofthequantitiesdiscussedinthepre- corresponding to a flat rotation curve. The non-
vious subsection. Data for x0, y0 and z0 are taken from Stanek axisymmetric potential due to the galactic bar is modeled
etal.(1997).ThequantityM/ρ0 containsthecentraldensityρ0. as a purequadrupole
1
Φ(2) = Q v2 cos[2(φ Ω t)] X ,
Model x0 y0 z0 b conv 2 T c − b × Φ(b2)conv
[pc] [pc] [pc]
r 3
X = b , r>r ,
G1 1478 754 611 Φ(b2)conv r b
G3 4537 2303 1725 (cid:16) (cid:17) r 3
EE22((21)) 11000000 550050 348080 XΦ(b2)conv = (cid:20)2−(cid:16)rb(cid:17) (cid:21) , r6rb , (39)
P1 1565 783 582 where r = 3.44 kpc, v = 240 km s−1, 0.1 < Q < 0.4
b c T
P2 2653 1320 975
(see also Minchev & Famaey 2010) and the spiral potential
P3 1869 939 713
is given by
E1 1650 868 474
G2 1298 694 572 Φ = ǫ cos[α˜ ln(r/r ) m(φ Ω t)], (40)
s conv s 0 s
E3 1075 563 442 − −
whereǫ isthespiralstructurestrength,0.015< ǫ <0.072
s s
| |
foratwo-armedspiralstructureand0.007< ǫ <0.031for
Model Q22/Q11 Q11/M M/ρ0 | s|
[–] [pc2 ] [pc3 ] afour-armedspiralstructure,ǫs=sign( α˜) ǫs ,α˜= 4
− ×| | −
and 8 for the two-armed and four-armed spiral structure,
G1 −0.4146 3.4271×106 1.0724×1010 −
respectively,wherethenegativesigncorrespondstotrailing
G3 −0.3938 4.7727×106 –
spirals,misanintegercorrespondingtothenumberofarms,
E2(1) −0.4151 6.3600×106 5.0266×109 Ω 0.7,0.8,0.9,1.0,1.1 v /R , i.e., Ω [km s−1 kpc−1]
E2(2) −0.4017 6.3777×106 4.9245×109 s 2∈1,{24,27,30,33 ,if v }=×24c0 km0 s−1 ansd R =8.0 kpc.
P1 −0.3957 – – ∈{ } c 0
P2 −0.3957 – –
P3 −0.3999 – –
E1 −0.3224 8.9338×106 5.4309×109 4.2 Physical approach
G2 −0.4461 5.7115×106 1.0639×1010
In this section we take the background axisymmetric po-
E3 −0.4294 1.2084×107 6.9228×109
tential due to the disk and halo in the form similar to Eq.
(38)
Φ = v2 1 ln(r/r ) , (41)
Table2.NumericalvaluesofquantitiesforthemodelsP1,P2and disk+halo − c { − G }
P3.ThevaluesofQ11/M andM/ρ0 holdforrb=3.44kpc.The where rG is the dark matter radius of the Galaxy. Eq. (41)
relative errors ̺(Q11/M) and ̺(M/ρ0) hold for the uncertainty is the physical potential corresponding to the flat rotation
of20%inrb. curve defined by the condition v(r) = v . The result pre-
c
sented byMinchev & Famaey (2010) or Dehnen(2000), see
Model Q22/Q11 Q11/M ̺(Q11/M) our Eq.(38), differs from Eq. (41).
[–] [pc2 ] [%]
The potential due to the galactic bar is given by Eqs.
P1 −0.3957 2.3149×106 −31/ +34 (3) and (10):
P2 −0.3957 2.1628×106 −32/ +36
P3 −0.3999 2.5152×106 −29/ +31 Φbar(~r) = Φ(0)(~r) + Φ(2)(~r) ,
1 G M
Φ(0)(~r) = G ̺(~r ′) d3~r ′ = ,
Model Q22/Q11 M/ρ0 ̺(M/ρ0) − r − r
[–] [pc3 ] [%] GQ Z(1 Q /Q )
Φ(2)(x,y,z) = 11 − 22 11
P1 −0.3957 0.9700×109 −20/ +17 − 4 r5
P2 −0.3957 6.8383×109 −19/ +16 1+Q /Q
P3 −0.3999 5.1389×109 −22/ +18 × 1 Q2222/Q1111 x2+y2−2 z2
(cid:20) −
+ x2 y2 cos((cid:0)2α)+2xysin(2(cid:1)α) ,
−
4.1 Conventional approach M = 1.4(cid:0) 1010M(cid:1)⊙ , (cid:3)
×
The conventional approach to galactic motion of the Sun Q11[M⊙ pc2] = 8×106 ×M[M⊙] ,
defines the following potentials of the Galaxy (Minchev et Q22/Q11 = 2/5 ,
−
al. 2010). The background axisymmetric potential due to α = Ω t + α ,
b 0
the disk and halo has theform α =. 25◦ ,
0
−
Φ = v2 ln(r/r ) (38) Ω = 55.5 km s−1 kpc−1 , (42)
0 conv c 0 b −
6 J. Klaˇcka et al.
where also the value of M from Dauphole & Colin (1995 - Table 3. Numerical values of minimum and maximum galacto-
Table 5) is used and the valuesfrom Table 1 are taken into centricdistancesoftheSunundertheactionofgalacticbar,disk
account. andhalo.Motionforthetimespanof5×109yearsisconsidered.
Thespiralpotentialisgivenbytheconventionalmodel
represented by Eq. (40), in a slightly corrected form (Rb = vc vy0 Rmin Rmax
3.44 kpc): [kms−1 ] [kms−1 ] [kpc] [kpc]
Φ = ǫ v2cos[α˜ ln( R/R ) m(φ φ +Ω t)] 240.000 −255.196 7.384 9.193
s corr s c b − − s0 s
R 202.176 −220.000 7.774 8.317
= ǫ v2 S cos mφ +α˜ln
s c 1 s0 R
b
n (cid:16) R (cid:17)
+ S sin mφ +α˜ln ,
2 s0 R for Eqs.(41, 42, 43) the first case ,
b −
S1 = cos(mφ)co(cid:16)s(mΩst)−sin(m(cid:17)φo)sin(mΩst) , vy = vy0 =− 220.0 km s−1 ,
S2 = sin(mφ)cos(mΩst)+cos(mφ)sin(mΩst) , for Eqs.(41, 42, 43) the second case ,
−
x 4 y 4 v = v =+ 240.0 km s−1 ,
cos(4φ) = 4 + 3 , y y0
R R − for Eqs.(38, 39, 40) , (45)
(cid:20)(cid:16) (cid:17) (cid:16) (cid:17) (cid:21)
x y x 2 y 2
sin(4φ) = 4 , where thevalue of vy follows from the relations
R R R − R
(cid:20)(cid:16) (cid:17) (cid:16) (cid:17) (cid:21) v = v2+Σ(v2) ,
x 2 y 2 y − c
cos(2φ) = R − R , Σ(v2) = [vp(R0)]2 Φ(0) + [v(R0)]2 Φ(2) ,
(cid:16) x(cid:17)y (cid:16) (cid:17)
sin(2φ) = 2 R R , [v(R )]2 = (cid:0)R ∂Φ(0)(cid:1) =(cid:0) GM , (cid:1)
0 Φ(0) ∂R R0
(cid:20) (cid:21)R0
R = x2+y2 , (cid:0) (cid:1) ∂ G Q Q
[v(R )]2 = R 11 1+ 22
φs0 = αp0 , (43) 0 Φ(2) ∂R − 4R3 Q11
(cid:20) (cid:26) (cid:18) (cid:19)(cid:27)(cid:21)R0
(cid:0) (cid:1)
where the quantity φs0 denotes orientation of spiral arms = 3 1+ Q22 GM , (46)
in the coordinates x-y at the intial time t = 0 and we put 32 Q11 × R0
φ = α . The quantity ǫ is the spiral structure strength, (cid:18) (cid:19)
s0 0 s
0.015 < ǫ < 0.072 for a two-armed spiral structure and forthephysicalapproach.Eqs.(38)-(45)uniquelydetermine
s
0.007 < |ǫ |< 0.031 for a four-armed spiral structure, ǫ = motion of theSun in theGalaxy.
s s
| |
sign(α˜) ǫ ,α˜=+4and+8forthetwo-armedandfour-
s
×| |
armed spiral structure,respectively,wherethepositivesign
correspondstotrailingspirals,misanintegercorresponding 5 COMPUTATIONAL RESULTS
tothenumberofarms,Ω 0.7,0.8,0.9,1.0,1.1 v /R ,
s y0 0
∈{ }× We numerically solved equation of motion in accordance
where v is the initial circular velocity of the Sun and R
y0 0
with Eq.(44) and theresults presented in theprevioussec-
= 8.0 kpc.
tion.Weusedbothofthephysicalvaluesofv ,asitisgiven
y
by Eqs. (45 – thefirst and thesecond cases).
4.3 Equation of motion
If an object moves in theGalaxy undertheaction of galac- 5.1 Effect of the galactic bar
tic disk, halo, galactic bar and spiral structure, then the
Table 3 shows the extremal values of the evolution of the
equation of motion of theobject reads
galactocentricdistanceoftheSunundertheactionofgalac-
~v˙ = grad(Φdisk+halo + Φbar + Φs corr) , (44) tichalo,diskandgalacticbar.TheresultspresentedinTable
−
3 hold for the special case when the effect of spiral struc-
where Eqs. (41), (42) and (43) can be used.
tureisignored.Whilethenonexistenceofthenonsymmetric
parts in term Φ(2) would produce circular orbit for the ini-
4.4 Motion of the Sun – initial conditions tial conditions given by Eqs. (45 - the first and the second
cases),R=R =8kpc,thenonsymmetricpartsintheterm
0
Initial conditions for the Sunare Φ(2) cause slight oscillations in R(t).
x = R =8 kpc , TherelativechangeinRisabout15%(orevenlarger)
0
accordingtoFamaey&Minchev(2010),Minchev&Famaey
y = 0 ,
(2010). Our physical approach yields two different values.
vx = 0 , The first case, when vy0 = 255.196 km s−1, the initial
v = v = 255.2 km s−1 , orbital velocity of the Sun, −yields R /R = 0.923 and
y y0 min 0
−
Radial migration of the Sun in galactic disk 7
Table 4. Numerical values of minimum and maximum galacto- Table 5. Numerical values of minimum and maximum galacto-
centricdistancesoftheSunundertheactionofgalacticbar,disk centricdistancesoftheSunundertheactionofgalacticbar,disk
andhalo.Motionforthetimespanof5×109yearsisconsidered, + halo and spiral arms. The two-armed spiral structure and vc
vy0 =−220.0kms−1. =202.176kms−1,vy0=−220.0kms−1 andvx0 =0.0kms−1
are considered. The results hold for the time integration of 5 ×
Q11/M vc Rmin Rmax 109 years. The Sun remains (T / 100) × 5 × 109 years in the
[pc2 ] [kms−1 ] [kpc] [kpc] galactocentric distances R∈h7.8, 8.2ikpc.
3.4271×106 202.742 7.752 8.300
1.2048×107 201.669 7.796 8.333 ǫs Ωs Rmin Rmax T
[−] [vy0/R0 ] [kpc] [kpc] [%]
0.7 7.693 8.266 84.9
0.8 7.759 8.225 93.3
R /R = 1.149. While thevalue of R corresponds to
max 0 max 0.015 0.9 7.749 8.223 93.1
the value given by Famaey & Minchev (2010), the value of
1.0 7.758 8.232 94.5
Rmin is greater than the value presented by other authors. 1.1 7.744 8.224 92.8
Moreover,wehavetostressthattheeffectofspiralarmscon-
sidered by Minchev & Famaey (2010), Famaey & Minchev 0.7 7.434 8.445 52.7
0.8 7.632 8.248 76.5
(2010), Minchev et al. (2010), Minchev et al. (2011) is not
0.030 0.9 7.673 8.197 82.6
considered in our calculations. The second case, when v
y0 1.0 7.658 8.141 74.9
= 220.0 km s−1, yields R /R = 0.972 and R /R
− min 0 max 0 1.1 7.696 8.164 85.4
= 1.040. The galactocentric distance of the Sun changes in
much smaller intervalthan it was presented in theprevious 0.7 7.109 8.752 39.0
papers. Again, the effect of spiral arms is not considered in 0.8 7.434 8.362 51.1
0.045 0.9 7.519 8.298 60.4
our calculations.
LetusconsidermotionoftheSunduring5 109 years. 1.0 7.509 8.225 59.3
Thefirst case, characterized bythevaluev =2×40 km s−1, 1.1 7.582 8.200 67.5
c
yields that Sun moves within galactocentric distance R 0.7 6.772 8.971 29.3
∈
7.75 kpc,8.25 kpc during 44.3% of the total integration 0.8 7.201 8.534 37.0
htime, and, R 7.i5 kpc,8.5 kpc during 71.5% of the to- 0.060 0.9 7.339 8.459 44.2
tal time. The∈sechond case, charaicterized by v = 220.0 1.0 7.333 8.346 44.3
y0
km s−1, yields that Sun moves within galactocentr−ic dis- 1.1 7.437 8.268 50.0
tanceR 7.9kpc,8.1kpc during50.5% ofthetotaltime,
∈h i
and, R 7.8 kpc,8.2 kpc during 83.2% of the total time.
∈ h i
Theresultsdonotchangeifweconsiderthetotaltimespan
10 109 years instead of 5 109 years.
structure strength. The found interval for radial migration
× ×
In reality, there exists some dispersion in the values of
of the Sun (7.8 8.4) kpc is much shorter than the result
Q .TakingintoaccounttheresultsofTable1,wecancon- −
11 presentedbyMinchev&Famaey(2010),Famaey&Minchev
siderthefollowing twoextremalvalues:Q /M =3.4271
11 (2010),Minchevetal.(2010),Minchevetal.(2011).Wecan
106 pc2 for the model G1, and, Q /M = 1.2048 107 pc×2
11 explainthedifferencebyincorrectnessoftheequationsused
×
forthemodelE3.Theresultsfortheradialmigrationofthe
by the above mentioned authors. Moreover, it seems to be
SunaresummarizedinTable4.TheTable4showsthatthe
nonphysicaltousethecaseofresonanceΩ =v /R asthe
s y0 0
values are: R > 7.75 kpc and R 68.33 kpc.
min max relevant for the motion of the Sun in the Galaxy. There is
noargumentwhythespiralarmsshouldrotatewiththean-
gular velocity equalto theangular velocity of theSun.The
5.2 Simultaneous effect of the galactic bar and
well-known result statesthat aflatrotation curveexistsfor
the spiral structure
the Galaxy. The flat rotation curve yields an orbital speed
If we take into account Eqs. (43), then galactic radial mi- practically independent on the galactocentric distance and
gration of the Sun is less relevant than Minchev & Famaey theangularvelocityisgivenbytheratiooftheorbitalspeed
(2010), Famaey & Minchev (2010), Minchev et al. (2010), and the galactocentric distance. Thus, theangular speed of
Minchev et al. (2011) present. This holds both for the two- agalactic object isafunctionof thegalactocentric distance
and four-armed spiral structures, although the effect of the and the fixed value of Ωs, Ωs = vy0/R0, is a pure mathe-
two-armed spiral structure is much more relevant than the matical toy and it contains norelevant physics.
effectofthefour-armedspiralstructure,seeTables5and7. ThelastcolumninTables5-7denotestherelativetime
The results summarized in Tables 6 and 7 show that T oftheexistenceoftheSuninthegalactocentric distances
radial migration of the Sun is practically unaffected by the R 7.8, 8.2 kpc. The two-armed spiral structure yields
∈ h i
effect of the four-armed spiral structure with v = 202.176 T < 50% for large values of spiral structure strength. The
c
kms−1.Theresultsarepracticallyindependentonthespiral four-armed spiral structureyields T (69%, 79%).
∈
8 J. Klaˇcka et al.
Table 6. Numerical values of minimum and maximum galacto- Table 7. Numerical values of minimum and maximum galacto-
centricdistancesoftheSunundertheactionofgalacticbar,disk centricdistancesoftheSunundertheactionofgalacticbar,disk
+ halo and spiral arms. The four-armed spiral structure, vc = + halo and spiral arms. The four-armed spiral structure, vc =
202.176 km s−1, vy0 = −220.0 km s−1 and vx0 = 0.0 km s−1 202.176 km s−1, vy0 = −220.0 km s−1 and vx0 = 0.0 km s−1
are considered. The results hold for the time integration of 1 × are considered. The results hold for the time integration of 5 ×
109 years. The Sun remains (T / 100) × 1 × 109 years in the 109 years. The Sun remains (T / 100) × 5 × 109 years in the
galactocentric distances R∈h7.8, 8.2ikpc. galactocentric distances R∈h7.8, 8.2ikpc.
ǫs Ωs Rmin Rmax T ǫs Ωs Rmin Rmax T
[−] [vy0/R0 ] [kpc] [kpc] [%] [−] [vy0/R0 ] [kpc] [kpc] [%]
0.7 7.769 8.297 77.6 0.7 7.769 8.343 79.9
0.8 7.774 8.311 78.6 0.8 7.769 8.339 80.2
0.007 0.9 7.769 8.308 78.5 0.007 0.9 7.770 8.338 79.9
1.0 7.779 8.300 78.2 1.0 7.776 8.340 79.8
1.1 7.772 8.310 78.8 1.1 7.772 8.333 80.2
0.7 7.764 8.299 72.4 0.7 7.763 8.372 78.6
0.8 7.769 8.328 75.4 0.8 7.769 8.366 79.6
0.014 0.9 7.766 8.322 74.7 0.014 0.9 7.766 8.361 79.1
1.0 7.783 8.309 74.8 1.0 7.778 8.362 78.3
1.1 7.770 8.326 75.8 1.1 7.767 8.359 78.9
0.7 7.758 8.329 69.9 0.7 7.758 8.394 77.7
0.8 7.765 8.345 73.9 0.8 7.763 8.396 77.9
0.021 0.9 7.763 8.334 74.2 0.021 0.9 7.764 8.389 77.9
1.0 7.781 8.323 73.2 1.0 7.778 8.385 77.2
1.1 7.768 8.339 74.3 1.1 7.765 8.376 78.5
0.7 7.752 8.360 69.1 0.7 7.752 8.422 75.7
0.8 7.759 8.381 72.7 0.8 7.756 8.419 75.6
0.028 0.9 7.761 8.357 72.4 0.028 0.9 7.759 8.415 76.1
1.0 7.778 8.340 72.9 1.0 7.771 8.406 76.2
1.1 7.763 8.355 73.3 1.1 7.762 8.395 77.2
Table 8. Numerical values of minimum and maximum galacto-
6 DISCUSSION
centricdistancesoftheSunundertheactionofgalacticbar,disk
Tkmhesfi−r1s,tccoarsreesptroenadtesdtointhtehevaplrueeviuosuesdsbecyt,ioen.g,.,vcM=inch24ev0 += 2h0a2lo.1a7n6dkmspisra−l1aarmresc.oTnhsiedeforeudr-.aTrmheedresspuilrtaslhsotrlducftourrethaentdimvce
integration of 5 × 109 years. The initial x− component of the
& Famaey (2010), Famaey & Minchev (2010), Minchev et
al.(2010),Minchevetal.(2011).Thesecondapproachchar- Sun’s velocity is vx. The negative sign of vx corresponds to the
orientation toward the center of the Galaxy. The case when the
acterized by v = 220.0 km s−1 is the conventional ap-
y0 − resonancebetweentherotationoftheSunandspiralarmsiscon-
pbyroaIAchU,)itofustehsetShuenco2n2v0e.0ntkiomnasl−s1p.eeAds(ae.gc.o,nrseecqoumenmceen,dthede sRid0e=red8:kǫpsc=. 0.028, Ωs = 1.0 vy0/R0, vy0 = 220.0 km s−1 and
radial galactic migration of the Sun is smaller than 0.32
±
kpcduringtheinterval(5−10)×109years.Thus,theradial vx Rmin Rmax
migration found by Minchev & Famaey (2010), Famaey & [kms−1] [kpc] [kpc]
Minchev(2010),Minchevetal.(2010),Minchevetal.(2011)
is caused not only by an incorrect equation of motion, but +10.0 7.558 8.752
0.0 7.771 8.405
also bythefact that theeffect ofgalactic bartogetherwith
−10.0 7.774 8.429
othergalacticcomponentsissensitivetotherealvelocityof
the Sun.
Whichofthetwovaluesofv ismorerealistic?Wecan
y0
answerthisquestionbycalculatingtheOortconstants.The andB= 12.5kms−1kpc−1.Thefirstcaseisnotconsistent
firstcase,correspondingtov =240kms−1,yieldsA=17.1 with the−most probably values of the Oort constants, e.g.,
c
kms−1 kpc−1 andB = 14.9kms−1 kpc−1,ifthegalactic Olling&Merrifield(1998)yieldA=(11-15)kms−1 kpc−1,
disk, halo (Eq. 41) and−the dominant parts of the galactic B = -(12-14) km s−1 kpc−1 (seealso Lequeux2005, p.10).
bar (Eq. 42 – Φ(0)(~r), Φ(2)(~r)) are considered. The second Thus,thephysicalresultshouldcorrespondtov = 220.0
y0
case, v = 220.0 km s−1 yields A = 15.0 km s−1 kpc−1 km s−1. Galactic migration of the Sun is less than− 0.4
y0
− ±
Radial migration of the Sun in galactic disk 9
Table 9. Numerical values of minimum and maximum galacto- REFERENCES
centricdistancesoftheSunundertheactionofgalacticbar,disk
Dauphole B., Colin J., 1995, Astron.Astrophys., 300, 117
+ halo and spiral arms. The two-armed spiral structure and vc Dehnen W., 2000, Astrophys.J., 119, 800
= 202.176 km s−1 are considered. The results holdfor the time
integration of 5 × 109 years. The initial x− component of the FamaeyB.,MinchevI.,2010,InProceedingsoftheAnnual
Meeting of the French Society of Astronomy and Astro-
Sun’s velocity is vx. The negative sign of vx corresponds to the
orientation toward the center of the Galaxy. The case when the physics.S.Boissier, M.Heydari-Malayeri,R.Samadiand
resonancebetweentherotationoftheSunandspiralarmsiscon- D. Valls-Gabaud (eds.), Societe Francaise d’Astronomie
sidered:ǫs=0.06,Ωs=1.0vy0/R0,vy0=220.0kms−1andR0 et d’Astrophysique(SF2A),37-40.
=8kpc. Freeman K. C., 1970, Astrophys.J., 160, 811
Lequeux J., 2005, The Interstellar Medium. Springer-
vx Rmin Rmax Verlag, Berlin, 437 pp.
[kms−1] [kpc] [kpc] Majaess D.J., 2010, Acta Astronomica, 60, 55
Minchev I., Famaey B., 2010, Astrophys.J., 722, 112
+10.0 7.324 8.436
MinchevI.,Boily C.,SiebertA.,BienaymeO.,2010, Mon.
0.0 7.333 8.346
Not. R.Astron. Soc., 407, 2122
−10.0 7.120 8.667
Minchev I., Famaey B., Combes F., Di Matteo P.,
MouhcineM.,WozniakH.,2011,Astron.Astrophys.,527,
A147
OllingR.P.,Merrifield M.R.,1998, Mon.Not.R.Astron.
kpc with respect to the current value 8.0 kpc for the four- Soc. 297, 943
armed spiral structure. If we would add a nonzero value of Stanek K. Z., Udalski A., Szymanski M., Kaluzny J., Ku-
radial velocity component to the velocity of the Sun, then biak M., Mateo M., Krzeminski W., 1997, Astrophys. J.
the radial migration might be more relevant, see Table 8. 477, 163
However,thetwo-armedspiralstructureyieldsmuchgreater
radial migration, up to 1.2 kpc for the largest value of the
spiral structure strength. If we would add a nonzero value
ofradialvelocitycomponenttothevelocityoftheSun,then
the radial migration might be more relevant,see Table 9.
7 CONCLUSION
Thepaperpresentsdetailedderivationoftherelevantequa-
tions of motion for theSunundertheaction of galactic bar
and spiral arms. The equations differ from the equations
used byother authors.
Improving physics presented by Dehnen (2000),
Minchev & Famaey (2010), Famaey & Minchev (2010),
Minchev et al. (2010), Minchev et al. (2011) and others,
we have obtained significantly different radial migration of
the Sunthan the previousauthors.
Galactic migration of theSun is less than 0.4 kpc, if
±
thefour-armed spiral structureand thevaluev = 220.0
y0
km s−1 are considered. The Sun remains (69 7−9)% of
−
its existence within the galactocentric distances (7.8 8.2)
−
kpc and the result is practically insensitive on the spiral
structure strength. Thus, the radial distance changes only
within 5% from thevalueof 8 kpc.
Galactic radialmigration oftheSunisless than (0.3
±
1.2) kpc for the two-armed spiral structure and v =
y0
−220.0 km s−1. The Sun remains (29 95)% of its exis-
− −
tence within galactocentric distances (7.8 8.2) kpc and
−
theresults strongly depend on thespiral structurestrength
andtheangularspeedofthespiralarms.Theradialdistance
changes within (3.8 15.0)% from thevalueof 8 kpc.
−