hydrodynamical escape approximation for plutoeanam6.khu.ac.kr/presentations/7-1.pdfa violent...

Hydrodynamical Escape Approximation for Pluto

David, Chien-Chang YenMathematics, Fu Jen Catholic University, Taiwan

Danie, Mao-Chang LiangResearch Center for Environmental Changes, Academia Sinica,

TaiwanRonald E. Taam

Astronomy & Atrophysics, Academia Sinica, Taiwan

The 6the East-Asian Numerical Astrophysics MeetingKyung Hee University, Suwon, Korea

September 18, 2014

David, Chien-Chang Yen Mathematics, Fu Jen Catholic University, Taiwan Danie, Mao-Chang Liang Research Center for EnvironmentalHydrodynamical Escape Approximation for Pluto

Outline

◮ Hydrodynamic Escape and the Corresponding Equations

◮ Boundary Conditions

◮ Relaxation Methods

◮ Previous Results

◮ Our Numerical Results

◮ A Comparison Study

◮ An Application on Isotope

◮ Conclusion


HD 209458

Artist’s illustration of the hot Jupiter-like planet orbiting star HD209458. Astronomers have now discovered oxygen and carbon in ina violent atmospheric ’blow-off.’ (Image credit: AlfredVidal-Madjar)


Two Classes of Escape MechanismThermal or Non-Thermal


UV photon heat the upper atmosphere


Absorption of Solar EUV/UV

Solar UV absorption drives atmospheric:

◮ constituent densities,

◮ thermal structure, and

◮ dynamics

Solar UV is absorbed by

◮ ozone (200-320 nm)

◮ molecular oxygen (140-242 nm)


Isothermal Atmosphere (Hydrostatic)

dp

dz= −ρg

p = nkT = ρk

mT, ρ = mp/kT

dp

p= −

mg

kTdz ≡ −

dz

Ha

ln p = −z

Ha+ const.

p = p0 exp

(

−z − z0

Ha

)

As z → ∞, p → 0 as expected.


Parker Isothermal Solution

u = cs, r = rs =GMs

2c2s

(u

)2 − ln(u

)2 − 4 ln(r

) − 4rs

= C = Const.David, Chien-Chang Yen Mathematics, Fu Jen Catholic University, Taiwan Danie, Mao-Chang Liang Research Center for EnvironmentalHydrodynamical Escape Approximation for Pluto

Gravity Varies with Height

g =GM

r2

dp

p= −

GMm

r2kTdr

ln p =GMm

rkT+ const.

p = p0 exp(GMm

kT(1

r−

1

r0)).

As r → ∞, p → p0 > 0 ⇒ infinite mass.


Modifications

◮ The atmosphere becomes collisionless at some height, so thatpressure is not defined in the normal manner.

◮ The atmosphere is not hydrostatic, i.e. it must expand into space.


Hydrodynamic Escape Equations

∂(ρr2)

∂t+

∂(ρur2)

∂r= 0, (1)

∂(ρur2)

∂t+

∂(ρu2r2 + pr2)

∂r= −ρGM + 2pr, (2)

∂(Er2)

∂t+

∂((E + p)ur2)

∂r= −ρuGM + qr2 +

∂

∂r(κr2 ∂T

∂r), (3)

where the total energy E, internal energy e, and the pressure p are

given by E = ρ(u2

2 + e), e = pρ(γ−1) , p = ρRT. Here, heating

(q) from the central star and thermal conduction (coefficientκ = νT are also imposed, where ν is a constant) in theatmosphere are also considered.


A Mathematical Problem

du

dr=

u

u2 − γkBT/µme× [2γkBT/(µmer) − (γ − 1)q/(ρu)

−GMp

r2−

γ − 1

ρur2

d

dr(κr2 dT

dr)

]

(4)

dρ

dr= −

ρ

u

du

dr−

2ρ

r(5)

dT

dr= (γ − 1)

(

q

ρu

µme

kB+

T

ρ

dρ

dr+

1

r2ρu

µme

kB

d

dr(κr2 dT

dr)

)

.(6)


Boundary Conditions

We impose the density ρb and the temperature Tb at the bottomboundary rb,

ρb = ρ(rb, t), Tb = T (rb, t). (7)

There two conditions are required. The transonic solution isdesired, and it follows that

u2 = c2 = γRT, (8)

and

2γRT/r − (γ − 1)q/(ρu) −GMp

r2−

γ − 1

ρur2

d

dr(κr2 dT

dr) = 0 (9)

at the sonic position rs.


More Boundary Conditions

We introduce three extra dependent variables

L = −κr2 dT

dr, W =

dL

dr, and Z = rs − rb.

L = −κr2(γ − 1)

[

T (−1

u

du

dr

∣

∣

rs

−2

r) −

Wµme

ρur2kB

]

, (10)

◮ The mathematical problem is well-posed.

◮ The existence and uniqueness of the solution of the HEP isun-resolved.


Relaxation Method

The quantities (T, u, ρ, Z,L,W ) are given at discrete points:

du

dr=

u

u2 − γkBT/µme×

[

2γkBT/(µmer) − (γ − 1)q/(ρu) −GMp

r2+

γ − 1

ρur2W

]

dρ

dr= −

ρ

u

du

dr−

2ρ

rdT

dr= −

L

κr2

dL

dr=

ρur2kB

µme

[

−1

γ − 1

dT

dr+

T

ρ

dρ

dr+

µme

ρukBq

]

dL

dr= W,

dZ

dr= 0


The Derivative of Velocity at the Sonic Position

du

dr|rs

=γ − 1

γ + 1

{

−2u

r−

γW

2ρu2r2+

1

2u

[

(γW

ρur2)2 − 8

(γ + 1)

(γ − 1)

uW

ρr3

+16

(γ − 1)

uW

ρr3+

8(5 − 3γ)

(γ − 1)2u4

r2

]1/2}

,


Darrell F. Strobel, 2008, Icarus

◮ The number density 4E+12 (1/cm3) and the temperature 88.2 Kat the lower boundary.

◮ Conduction κ(T ) = 8.37T and Ms = 1.305E + 25 g.


Results of Strobel


Results of the Relaxation Method


Hybrid Fluid/Kinetic Models

The hybrid fluid/kinetic model of Erwin, Tucker, and Johnson(2012) attempts to describe the atmospheric escape morerealistically. Such an approach consists of the hydrodynamicescape result of Strobel (2008) and a modified Maxwell-Boltzmannto include the bulk velocity u in the velocity distribution(Yell,2004; Tian et al., 2009; Volkov et al., 2011). However, thetreatment of the transition from the collisional regime to the noncollisional regime is approximate.


Jeans escape

Jeans escape corresponds to the literal evaporation atom by atom,molecule by molecule, off the top of the atmosphere. At loweraltitudes, collisions confine particles, but above a certain altitude,known as the exobase, air is so tenuous that gas particles hardlyever collide.The Jean’s escape framework in terms of the Jeans parameter,λ(r) = GMµmp/rkBT . The Jean’s escape rate is given as

ΦJ = 4πr2xnx

√

kTx

2πµmp(1 + λx) exp(−λx).


Settings for Pluto Simulations

◮ The lower boundary rmin = 1.45E + 8 cm

◮ The temperature T = 88.2 at the lower boundary

◮ The number density n = 4E + 12 at the lower boundary

◮ The mass of the Pluto Mp = 1.31E + 25 g

◮ For nitrogen simulation, µ = 28.

We request the heat profile q of solar mean from Strobel for twonumerical simulations.


Results for Pluto Simulations

4 6 8 10 12

1

2

3

4

5

6

x 104 log

10(ρ

n)

0 2000 4000 6000 8000

1

2

3

4

5

6

x 104 u

20 40 60 80 100 120 140

1

2

3

4

5

6

x 104 T

40 42 44 46

1

2

3

4

5

6

x 104 z

−2000 0 2000

1

2

3

4

5

6

x 104 L

−1 0 1

x 104

1

2

3

4

5

6

x 104 W


Models

◮ Model A is a consistent model.

◮ Model B is a fixed heat profile in Strobel (2009) to compare withStrobel’s result.

◮ Model C is a fixed heat profile to compare with Johnson’ssimulations.


Comparision

Model Tpk r(Tpk) M Qunit K 108 cm 1027cm−3 erg/sA 137 2.20 3.98 16.06B 132 2.08 1.73 11.35C 123 2.07 0.77 7.8

Strobel 117 2.02 3.26 11.35Johnson 120 2.13 2.58 7.8

This table shows us the peak temperature Tpk, its location r(Tpk),

the mass flux M , and the total heat Q for the model A, B, C,Strobel and Johnson’s simulations.


Comparison

Model r(Texo) ρ Texo λ(r(Texo))unit 108 cm 108cm−3 KA 7.93 0.014 67 5.6B 7.20 0.015 68 6.2C 6.20 0.014 71 6.7

Strobel 4.83 0.027 66 9.3Johnson 7.40 0.002 80 3.8

This table shows us the exobases for temperature and density andtheir corresponding temperature, density and the Jeans parameterfor Model A, B, C, Strobel and Johnson.


Comparison

Model r(ρexo) ρexo T λ(r(ρexo))unit 108cm−3 KA 3.5 1.55 112 7.5B 3.4 1.62 106 8.2C 3.2 1.86 105 9.0

Strobel 3.0 1.97 95 10.4Johnson 9.7 3.11 74 4.1

This table shows us the exobases for temperature and density andtheir corresponding temperature, density and the Jeans parameterfor Model A, B, C, Strobel and Johnson.


Comparison of results for Pluto’s atmosphere

40 60 80 100 120 1401000

2000

3000

4000

5000

6000

7000

8000

9000

10000

x

o

x

o

x

o

104 106 108 1010 1012 1014 1016 1018 1020 1022 1024

40 60 80 100 120 1401000

2000

3000

4000

5000

6000

7000

8000

9000

10000 xo

x

o

x

o

104 106 108 1010 1012 1014 1016 1018 1020 1022 1024

The superpose profiles of density and temperature for the modelsA, B, Strobel, and A, C, Johnson labeled by red, blue and blackline are shown in left and right, respectively. The marker “x” and“o” indicate the exobase locations.


New Settings for Pluto Simulations

◮ The lower boundary rmin = 1.18E + 8 cm

◮ The temperature T = 37 at the lower boundary

◮ The number density T = 4E + 15 at the lower boundary

◮ The mass of the Pluto Mp = 1.31E + 25 g

◮ For nitrogen simulation, µ = 28.

We request the heat profile q of solar mean from Strobel for twonumerical simulations.


Comparison of results for Pluto’s atmosphere

1 2 3 4 5 6 7 8 9 10

40

60

80

100

120

140

tem

pera

ture

(K

)

r/r0

1 2 3 4 5 6 7 8 9 100

5

10

15

log 10

(ρn)

r/r0


Notations for isotope simulations

Let us denote the density, velocity, and temperature for the majorspecies as ρ, u, and T , respectively; the corresponding quantitiesfor the minor constituent are labeled with the subscript “i.” Thetemperature T , the velocity u, and the density ρ are fixed for thesimulation of the minor species.


Equations for isotope simulations

Following (Zahnle and Kasting 1986), they are described as

d

dr(ρiuir

2) = 0, (11)

d

dr(ρiu

2i r

2 + pir2) = −GMpρi + 2pir

+ ρ2νi(u − ui)r2Φi (12)

where Φi = 1, νi = σnc = σn√

γkBT/µmp andpi = ρikBT/µimp.


Approximation of isotope solutions

Under the assumption of the temperature of the major and minorspecies is the same, T = Ti, the mass and momentum equationsfor minor species are only needed to be solved. Equations (11) and(12) can be rewritten as

dui

dr=

ui

u2i − kBT/µimp

×

[

−kB

µimp

dT

dr+

2kBT

µimpr−

GMp

r2+ νi(u − ui)Φi

]

(13)

dρi

dr= −

ρi

ui

dui

dr−

2ρi

r(14)

◮ The singularity occurs at the thermal critical point.


Thermal critical

Since the solution is required to be smooth and (13), the velocityui at the singular position rc should satisfy

u2i =

kBT

µimp(15)

and

−kB

µimp

dT

dr+

2kBT

µimprc−

GMp

r2+ νi(u − ui)Φi = 0. (16)

In the other words, the numerator and denomerator in (13) arevanished. This singular position rc is identified as a thermal criticaland also regarded as the top boundary.


Results for isotope simulation

0 5 10 15 203

3.5

4

4.5

5

ρn

1

(dashed); ρn

2

(solid)

log10

(ρn)

log 10

(km

)

−8 −6 −4 −2 0 2 43

3.5

4

4.5

5

u1(dashed);u

2(solid)

log10

(cm/sec)

0 100 200 3003

3.5

4

4.5

5

ρn

1

/ρn

2

log 10

(km

)

1 2 3 4 53

3.5

4

4.5

5

u1/u

2

M = 1.4702 × 1026cm−3 and Mi = 5.7139 × 1023cm−3


Conclusion

◮ The mathematical study of the hydrodynamic escape problem isin its infancy, since even the existence and uniqueness of solutionsare still open. On the other hand, through our numericalsimulations, the hydrodynamic solution can approximate thesituation of the rare gas for Pluto.

◮ Under the framework of hydrodynamic approximation, it wasgenerally accepted that the process produced rather small isotopicfractionation. Here, we show that the escape highly fractionatesthe isotopic composition of nitrogen.


Thanks for your attention


References

◮ J. Erwin, O. J. Tucker, R. E. Johnson, Hybrid fluid/kineticmodeling of Pluto’s escaping atmosphere, Icarus, 226, 375-384,2013.

◮ T. E. Holzer and W. I. Axford, The Theory of Stellar Wind andRelated Flows, Annual Review of Astronomy and Astrophysics, 8,31-60, 1970.

◮ J. M. Hong, C. C. Yen, and B. C. Huang, Characterization of theTransonic Stationary Solutions of the Hydrodynamic EscapeProblem, submitted to SIAM Journal on Applied Mathematics.


References

◮ M. C. Liang, R. E. Taam, David C. C. Yen, On the Suitability ofthe Hydrodynamical Approximation for Atmospheric Escape fromPluto, in preparation.

◮ R. A. Murray-Clay, E. I. Chiang, and N. Murray, AtmosphericEscape From Hot Jupiters, The Astrophysical Journal, 693,23-43, 2009.

◮ Darrell F. Strobel, N2 escape rates from Pluto’s atmosphere,Icarus, 193, 612-619, 2008.

◮ F. Tian and O.B. Toon, Hydrodynamic escape of nitrogen fromPluto, Geophys. Res. Lett., 32, L1820, 2005.


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