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❆♣♣❧✐❝❛t✐♦♥ ♦❢ ●❛s ❉②♥❛♠✐❝❛❧ ❋r✐❝t✐♦♥ t♦ P❧❛♥❡t❡s✐♠❛❧s

❊✈❣❡♥✐ ●r✐s❤✐♥ ✫ ❍❛❣❛✐ ❇✳ P❡r❡ts✶

▲✉♥❞ ❖❜s❡r✈❛t♦r②✱ ▲✉♥❞✱ ❙✇❡❞❡♥ ❙✉♣♣♦rt❡❞ ❜② ❊✉r♦♣❡❛♥ ❋P✼❈❆● ❣r❛♥❞

❊①♦♣❧❛♥❡ts ✐♥ ▲✉♥❞✱ ✵✻✳✵✺✳✷✵✶✺

✶❚❡❝❤♥✐♦♥✱ ■sr❛❡❧ ■♥st✐t✉t❡ ♦❢ ❚❡❝❤♥♦❧♦❣②✱ ❍❛✐❢❛✱ ■sr❛❡❧

• ❛s ♣❧❛♥❡t❡s✐♠❛❧ ✐♥t❡r❛❝t✐♦♥ ♣❧❛② ❛♥ ✐♠♣♦rt❛♥t r♦❧❡ ✐♥ ♣❧❛♥❡t

❢♦r♠❛t✐♦♥

❢❡✇×✶✵✻②r ❞✐s❦ ❧✐❢❡t✐♠❡s ✭P❢❛❧③♥❡r✳✱ ✷✵✶✹✮

• ❛s ❞r❛❣ ✐s ✐♠♣♦rt❛♥t ❢♦r s♠❛❧❧

♣❧❛♥❡t❡s✐♠❛❧s ❚②♣❡ ■ ♠✐❣r❛t✐♦♥ ✐s ✐♠♣♦rt❛♥t ❢♦r ❧❛r❣❡ ♣❧❛♥❡t❛r② ❡♠❜r②♦s ❍♦✇ ❞♦❡s ❣❛s ❛✛❡❝t ✐♥t❡r♠❡❞✐❛t❡ ♠❛ss ♣❧❛♥❡t❡s✐♠❛❧s❄

❆❡r♦❞②♥❛♠✐❝ ❣❛s ❞r❛❣ ✐s ❡✛❡❝t✐✈❡ ❢♦r s♠❛❧❧ ♣❧❛♥❡s✐♠❛❧s

• ❛s ❉r❛❣ ❋♦r♠✉❧❛

❋❞ = −✶

✷❈❉(❘❡)❆ρ❣✈✷ r❡❧

❈❉ ✲ ❉r❛❣ ❝♦❡✣❝✐❡♥t ❘❡ ✲ ❘❡②♥♦❧❞s ♥✉♠❜❡r ❆ ✲ ❈r♦ss s❡❝t✐♦♥ ✈r❡❧ ✲ r❡❧❛t✐✈❡ ✈❡❧♦❝✐t②

❚✐❣❤t❧② ❝♦✉♣❧❡s s♠❛❧❧ ❣r❛✐♥s ■♥❡✣❝t✐✈❡ ❢♦r ❧❛r❣❡ ♣❧❛♥❡t❡s✐♠❛❧s ❑❡❡♣s r❡❧❛t✐✈❡ ✈❡❧♦❝✐t✐❡s ❧♦✇ ■♥❝r❡❛s❡s ❣r♦✇t❤

P❧❛♥❡t❛r② ♠✐❣r❛t✐♦♥ ✐s ❞♦♠✐♥❛♥t ❢♦r ❧❛r❣❡ ♣r♦t♦♣❧❛♥❡ts

✭▼❛ss❡t✳✱ ✶✾✾✾✮ P❧❛♥❡t❛r② ▼✐❣r❛t✐♦♥ ❊①❝❤❛♥❣❡ ♦❢ ❛♥❣✉❧❛r ♠♦♠❡♥t✉♠ ✇✐t❤ t❤❡ ❣❛s ✭▲✐♥ ✫ P❛♣❛❧♦✐③♦✉✱ ✶✾✼✾✮ ❙♣✐r❛❧ ❞❡♥s✐t② ✇❛✈❡ ✭●♦❧❞r❡✐❝❤ ✫ ❚r❡♠❛✐♥❡✱ ✶✾✽✵✮

❘❡s♦♥❛♥t ▲✐♥❞❜❧❛❞ ❛♥❞ ❝♦r♦t❛t✐♦♥ t♦rq✉❡s ♠|Ω(r)−Ω♣(r)| = ±κ(r), ♠ ∈ Z

❊✛❡❝t✐✈❡ ❢♦r ♠❛ss❡s ♦❢ ♠ ✶✵✷✺❣ ✭❍♦✉r✐❣❛♥ ✫ ❲❛r❞✱ ✶✾✽✹❀ ❚❛❦❛♥❦❛ ✫ ■❞❛✱ ✶✾✾✾✮

❙❝❛❧✐♥❣ ✇✐t❤ P❧❛♥❡t❡s✐♠❛❧ ▼❛ss

❉②♥❛♠✐❝❛❧ ❋r✐❝t✐♦♥ ✭❉❋✮ ✐s ❛♥ ❡✛❡❝t✐✈❡ ❣r❛✈✐t❛t✐♦♥❛❧ ❞r❛❣ ♠❡❝❤❛♥✐s♠

❉❋ ✐s ❛ ▲♦ss ♦❢ ♠♦♠❡♥t✉♠ ♦❢ ❛ ♠❛ss✐✈❡ ♦❜❥❡❝t ✐♥ ❛ ❜❛❝❦❣r♦✉♥❞ ♠❡❞✐✉♠✱ ❜② ❝r❡❛t✐♥❣ ❛♥ ♦✈❡r✲❞❡♥s✐t② ❣r❛✈✐t❛t✐♦♥❛❧ ✇❛❦❡ ❈♦❧❧✐s✐♦♥❧❡ss s②st❡♠s ✭❈❤❛♥❞r❛s❡❦❤❛r✱ ✶✾✹✸✮

• ❛s❡♦✉s ♠❡❞✐✉♠ ✭❖str✐❦❡r✱ ✶✾✾✾✮
• r❛✈✐t❛t✐♦♥❛❧ ♣❡rt✉r❜❛t✐t✐♦♥ ♦♥ ✉♥✐❢♦r♠ ❣❛s❡♦✉s ♠❡❞✐✉♠✿

❈❛❧❝✉❧❛t❡ t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ✇❛❦❡ α(①,t) = ∆ρ(①,t)/ρ✵ ❈❛❧❝✉❧❛t❡ t❤❡ ❡✛❡❝t✐✈❡ ❢♦r❝❡ ❋●❉❋ =

ρ∇Φ❡①t❞✸r

❉②♥❛♠✐❝❛❧ ❋r✐❝t✐♦♥ ✭❉❋✮ ✐s ❛♥ ❡✛❡❝t✐✈❡ ❣r❛✈✐t❛t✐♦♥❛❧ ❞r❛❣ ♠❡❝❤❛♥✐s♠

❉❋ ✐s ❛ ▲♦ss ♦❢ ♠♦♠❡♥t✉♠ ♦❢ ❛ ♠❛ss✐✈❡ ♦❜❥❡❝t ✐♥ ❛ ❜❛❝❦❣r♦✉♥❞ ♠❡❞✐✉♠✱ ❜② ❝r❡❛t✐♥❣ ❛♥ ♦✈❡r✲❞❡♥s✐t② ❣r❛✈✐t❛t✐♦♥❛❧ ✇❛❦❡ ❈♦❧❧✐s✐♦♥❧❡ss s②st❡♠s ✭❈❤❛♥❞r❛s❡❦❤❛r✱ ✶✾✹✸✮

• ❛s❡♦✉s ♠❡❞✐✉♠ ✭❖str✐❦❡r✱ ✶✾✾✾✮
• r❛✈✐t❛t✐♦♥❛❧ ♣❡rt✉r❜❛t✐t✐♦♥ ♦♥ ✉♥✐❢♦r♠ ❣❛s❡♦✉s ♠❡❞✐✉♠✿

❈❛❧❝✉❧❛t❡ t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ✇❛❦❡ α(①,t) = ∆ρ(①,t)/ρ✵ ❈❛❧❝✉❧❛t❡ t❤❡ ❡✛❡❝t✐✈❡ ❢♦r❝❡ ❋●❉❋ =

ρ∇Φ❡①t❞✸r

❉②♥❛♠✐❝❛❧ ❋r✐❝t✐♦♥ ✐♥ ●❛s❡♦✉s ▼❡❞✐✉♠ ✭●❉❋✮

▲✐♥❡❛r ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r② ②✐❡❧❞s ❛♥ ♦✉t❣♦✐♥❣ ♣r❡ss✉r❡ ✇❛✈❡ ❙♦❧✈✐♥❣ ■♥❤♦♠♦❣❡♥♦✉s ✇❛✈❡ ❡q✉❛t✐♦♥ ✇✐t❤ r❡t❛r❞❡❞ ♣♦t❡♥t✐❛❧ P♦✐♥t ♠❛ss ♣❡rt✉r❜❡r

• ❉❋ ♣❡❛❦s ♥❡❛r ✈ ∼ ❝s

❋ = ❋✵ ×I (M ) ✇❤❡r❡ ❋✵ = ✹π● ✷▼✷ρ✵

❝✷

s

▼ ✲ ♦❜❥❡❝t ♠❛ss ❝s ✲ s♣❡❡❞ ♦❢ s♦✉♥❞

I (M ) =

✶ M ✷ ×

• ✶

✷ ❧♥

✶+M

✶−M

• −M

M < ✶

✶ ✷ ❧♥(✶−M −✷)+❧♥Λ

M > ✶ ❆♣♣r♦①✐♠❛t❡ ❢♦r♠✉❧❛✿ I (M ) =

• M /✸

M ≪ ✶ ❧♥Λ/M ✷ M ≫ ✶ Λ = r♠❛①/r♠✐♥ ✐s ❝❛❧❧❡❞ ❈♦✉❧♦♠❜ ❧♦❣❛r✐t❤♠

Pr❡✈✐♦✉s ✇♦r❦s ❝♦♥s✐❞❡r❡❞ ♦♥❧② ♠❛ss❡s ♦❢ ▼⊕

• ❉❋ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ♣❧❛♥❡t ❢♦r♠❛t✐♦♥✿

❱❡rt✐❝❛❧❧② ❛✈❡r❛❣❡❞✱ st❡❛❞② st❛t❡ ●❉❋ ✭▼✉t♦ ❡❧✳ ❛❧✳✱ ✷✵✶✶✮

• ❉❋ ❞♦♠✐♥❛♥t ❢♦r ❤✐❣❤❧② ❡❝❝❡♥tr✐❝ ♦r❜✐t

❉✐s❦ ♣❧❛♥❡t ✐♥t❡r❛❝t✐♦♥s ❢♦r ❤✐❣❤❧② ✐♥❝❧✐♥❡❞ ♦r❜✐ts ✭❘❡✐♥✳✱ ✷✵✶✷✮ ■♥t❡r❛❝t✐♦♥ ♦❢ ❛❝❝r❡t✐♥❣ ♣❧❛♥❡t ✭▲❡❡ ✫ ❙t❛❤❧❡r✳✱ ✷✵✶✷❀ ❈❛♥t♦ ❡t✳ ❛❧✳✱ ✷✵✶✷✮ ❙❡❝✉❧❛r ✐♥t❡r❛❝t✐♦♥ ♦❢ s❡❧❢ ❣r❛✈✐t❛t✐♥❣ ❞✐s❦ ✭❚❡②ss❛♥❞✐❡r ❡t✳ ❛❧✳✱ ✷✵✶✸✮ ❆❧❧ ❝♦♥s✐❞❡r ♠❛ss❡s ♦❢ ❢✉❧❧② ❡✈♦❧✈❡❞ ♣❧❛♥❡ts✱ ❛t ❧❡❛st ▼⊕

P♦✇❡r ❧❛✇ ❉✐s❦ ❙tr✉❝t✉r❡

Pr♦t♦♣❧❛♥❡t❛r② ❉✐s❦ ❙tr✉❝t✉r❡ ✭●♦❧❞r❡✐❝❤ ✫ ❈❤✐❛♥❣✳✱ ✶✾✾✼✮ ❘❛❞✐❛❧ ❙tr✉❝t✉r❡✿

❚❡♠♣❡r❛t✉r❡ ♣r♦✜❧❡✿ ❚❞✐s❦ ≈ ✶✷✵(❛/❆❯)−✸/✼❑ ❙♦✉♥❞ s♣❡❡❞ ❝s ≈ ✹.✼×✶✵✹(❛/❆❯)−✸/✶✹❝♠/s ❆s♣❡❝t r❛t✐♦ ❍✵ = ❤(❛)/❛ = ✵.✵✷✷(❛/❆❯)✷/✼ ❘❛❞✐❛❧ ❣❛s ❞❡♥s✐t②✿ ρ❣(❛) = ✸×✶✵−✾(❛/❆❯)−✶✻/✼❣/❝♠✸

❱❡rt✐❝❛❧ str✉❝t✉r❡✿

❱❡rt✐❝❛❧ ●❛s ❞❡♥s✐t②✿ ρ❣(❛✵,③) ∼ ρ❣(❛✵,✵)×❡①♣(−③✷/✷❤✷)❣/❝♠✸ ■s♦t❤❡r♠❛❧ ❞✐s❦

❘❡❧❛t✐✈❡ ✈❡❧♦❝✐t② ❞✉❡ t♦ ♣r❡ss✉r❡ ❣r❛❞✐❡♥ts✿

Pr❡ss✉r❡ ❣r❛❞✐❡♥t✿ P ∼ (❛/❆❯)−β ✇❤❡r❡ β = ✶✾/✼ ✈r❡❧ = |✈❑ −✈ϕ,❣❛s| ∼ β❍✷

✵✈❑ ≪ ❝s

P♦✇❡r ❧❛✇ ❉✐s❦ ❙tr✉❝t✉r❡

Pr♦t♦♣❧❛♥❡t❛r② ❉✐s❦ ❙tr✉❝t✉r❡ ✭●♦❧❞r❡✐❝❤ ✫ ❈❤✐❛♥❣✳✱ ✶✾✾✼✮ ❘❛❞✐❛❧ ❙tr✉❝t✉r❡✿

❚❡♠♣❡r❛t✉r❡ ♣r♦✜❧❡✿ ❚❞✐s❦ ≈ ✶✷✵(❛/❆❯)−✸/✼❑ ❙♦✉♥❞ s♣❡❡❞ ❝s ≈ ✹.✼×✶✵✹(❛/❆❯)−✸/✶✹❝♠/s ❆s♣❡❝t r❛t✐♦ ❍✵ = ❤(❛)/❛ = ✵.✵✷✷(❛/❆❯)✷/✼ ❘❛❞✐❛❧ ❣❛s ❞❡♥s✐t②✿ ρ❣(❛) = ✸×✶✵−✾(❛/❆❯)−✶✻/✼❣/❝♠✸

❱❡rt✐❝❛❧ str✉❝t✉r❡✿

❱❡rt✐❝❛❧ ●❛s ❞❡♥s✐t②✿ ρ❣(❛✵,③) ∼ ρ❣(❛✵,✵)×❡①♣(−③✷/✷❤✷)❣/❝♠✸ ■s♦t❤❡r♠❛❧ ❞✐s❦

❘❡❧❛t✐✈❡ ✈❡❧♦❝✐t② ❞✉❡ t♦ ♣r❡ss✉r❡ ❣r❛❞✐❡♥ts✿

Pr❡ss✉r❡ ❣r❛❞✐❡♥t✿ P ∼ (❛/❆❯)−β ✇❤❡r❡ β = ✶✾/✼ ✈r❡❧ = |✈❑ −✈ϕ,❣❛s| ∼ β❍✷

✵✈❑ ≪ ❝s

• ❉❋ ✐s str♦♥❣❡r t❤❛♥ ●❛s ❉r❛❣ ❢♦r ❘❛❞✐✐ ❘ ✷✵✵❦♠
• ❛s ❞r❛❣ s❝❛❧❡s ❛s ∼ ❘✷
• ❉❋ s❝❛❧❡s ❛s ∼ ❘✻

❈♦r♦❧❧❛r② ❊①✐sts ❛ ❝r✐t✐❝❛❧ ✈❛❧✉❡ ❘⋆(●,ρ♠,✈r❡❧,❘❡,M ) ✇❤❡r❡ ❜♦t❤ ❢♦r❝❡s ❛r❡ ❡q✉❛❧ r⋆ = ✵.✷✾ ❈❉(❘❡) I (M ) ✶/✹ ✈r❡❧

• ρ♠

• ❉❋ ✐s str♦♥❣❡r t❤❛♥ ●❛s ❉r❛❣ ❢♦r ❘❛❞✐✐ ❘ ✷✵✵❦♠
• ❛s ❞r❛❣ s❝❛❧❡s ❛s ∼ ❘✷
• ❉❋ s❝❛❧❡s ❛s ∼ ❘✻

❈♦r♦❧❧❛r② ❊①✐sts ❛ ❝r✐t✐❝❛❧ ✈❛❧✉❡ ❘⋆(●,ρ♠,✈r❡❧,❘❡,M ) ✇❤❡r❡ ❜♦t❤ ❢♦r❝❡s ❛r❡ ❡q✉❛❧ r⋆ = ✵.✷✾ ❈❉(❘❡) I (M ) ✶/✹ ✈r❡❧

• ρ♠

−1 1 2 3 4 100 200 500

log [a/AU] Crirical size [km] GDF Dominates Gas Drag Dominates

−1 1 2 3 0.5 1 1.5

log [a/AU] Mach Number

e=0 e=0.02 e=0.04 e=0.1

❙❝❛❧✐♥❣ ✇✐t❤ P❧❛♥❡t❡s✐♠❛❧ ▼❛ss

• ❉❋ ✐s ❞♦♠✐♥❛♥t ❢♦r ✐♠t❡r♠❡❞✐❛t❡ ♠❛ss ♣❧❛♥❡t❡s✐♠❛❧s

• ❉❋ ✐♥❞✉❝❡❞ ❞❛♠♣✐♥❣ t✐♠❡ ✐s ❝♦♠♣❛r❛❜❧❡ t♦ ❞✐s❦ ❧✐❢❡t✐♠❡

P❧❛♥❡t❡s✐♠❛❧ ✇✐t❤ ♦r❜✐t❛❧ ♣❛r❛♠❡t❡rs (❛,❡,■) ✉♥❞❡r ❞✐st✉r❜✐♥❣ ❢♦r❝❡ P❡rt✉r❜❛t✐♦♥ ❞✉❡ t♦ ❞✐st✉r❜✐♥❣ ❢♦r❝❡ ❋ = ❋rr +❋ϕϕ +❋③③ ♠♣ ❞❛ ❞t = ✷ ❛✸/✷

• ▼⋆(✶−❡✷)

[❋r❡ s✐♥❢ +❋ϕ(✶+❡ ❝♦s❢ )]

❋♦r ❝✐r❝✉❧❛r ♦r❜✐t t❤❡ ❙▼❆ ❞❛♠♣✐♥❣ t✐♠❡s❝❛❧❡ ✐s τ❛ = ❛/˙ ❛ ≈ ✶ ✹π ❍✷

✵❝✸ s

• ✷ρ❣♠♣

= ✸×

• ♠♣

✷·✶✵✷✺❣ −✶ ▼②r ❊❝❝❡♥tr✐❝✐t② ❛♥❞ ✐♥❝❧✐♥❛t✐♦♥ ❞❛♠♣✐♥❣ ✐s ∼ ✷−✸ ♦r❞❡rs ♦❢ ♠❛❣♥✐t✉❞❡ ❢❛st❡r τ❡ = ❡/˙ ❡ ∼ ❡τ❛ τ■ = ■/˙ ■ ∼ ✺❍✷

✵τ❛

❲❡ ✐♥t❡❣r❛t❡ ♥✉♠❡r✐❝❛❧❧② ✷✲❜♦❞② ♣r♦❜❧❡♠ ✇✐t❤ ❡①t❡r♥❛❧ ❢♦r❝❡

❘❡s✉❧ts✿ ❈♦♣❧❛♥❛r ♠❛ss✐✈❡ ❜♦❞✐❡s ✭♠ = ✶✵✷✺❣✮ ❞❛♠♣ (❛,❡) ✐♥ ❞✐s❦ ❧✐❢❡t✐♠❡s

1 2 3 0.2 0.4 0.6 0.8 1

Time [Myr] Semimajor axis [AU]

e=0 e=0.02 e=0.1 e=0.3 e=0.8 e=0.04 −3 −2 −1 0.2 0.4 0.6 0.8

log (Time/Myr) Eccentricity

−3 −2 −1 −2 −1 1 2

Time [Myrs] log (Mach)

❘❡s✉❧ts✿ ■♥❝❧✐♥❡❞ ♠❛ss✐✈❡ ❜♦❞✐❡s ✭♠ = ✶✵✷✺❣✮ ❞❛♠♣ (❛,❡,■) ✐♥ ❞✐s❦ ❧✐❢❡t✐♠❡s

0.2 0.4 0.6 0.8 1 0.8 0.85 0.9 0.95 1

Time [Myr] Semimajor axis [AU]

e=0 e=0.04 e=0.1 e=0.3 e=0.8 −3 −2 −1 0.2 0.4 0.6 0.8

log (Time/Myr) Eccentricity

−4 −3 −2 −1 −2 −1 1 2

Time [Myrs] log (Mach)

0.2 0.4 0.6 0.8 1 2 4 6 8

Time [Myr] Inclination

❘❡s✉❧ts✿ ❉❡♣❡♥❞❡♥❝❡ ♦♥ ❞✐s❦ s✉r❢❛❝❡ ❞❡♥s✐t② Σ❣ ∼ ❛−α

0.5 1 1.5 2 2.5 0.2 0.4 0.6 0.8 1

Time [Myr] Semimajor axis [AU]

α=1 α=1.5 α=2 e=0 e=0.1

10

−3

10

−2

0.02 0.04 0.06 0.08 0.1

Time [Myr] Eccentricity 2 4 6 8 10 x 10

−3

0.986 0.988 0.99 0.992 0.994 0.996 0.998 1

Time [Myr] Semimajor axis [AU]

10

−2

10 10

−2

10

−1

10

Time [Myr] Mach number

❙❝❛❧✐♥❣ ✇✐t❤ ♣❧❛♥❡t❡s✐♠❛❧❧ ♠❛ss τ ∼ ♠−✶

♣

▼❡r❣❡r t✐♠❡s❝❛❧❡ ❢♦r ❜✐♥❛r② ♣❧❛♥❡t❡s✐♠❛❧s ✐s s❤♦rt❡r

❋♦r ❝✐r❝✉❧❛r ❜✐♥❛r② t❤❡ ♠❡r❣❡r t✐♠❡s❝❛❧❡ ✐s✷ τ❛ = ❛❜✐♥/˙ ❛❜✐♥ ≈ ✸❝✸

s

✽π● ✷♠❜✐♥ρ❣ ∼ ❍✷

✵τ❛ ∼ ✵.✼✸

• ♠❜✐♥

✹·✶✵✷✸❣ −✶ ▼②r ❢♦r ❡❝❝❡♥tr✐❝ ♦r❜✐t ❛r♦✉♥❞ t❤❡ s✉♥✱ t❤❡ t♦rq✉❡ ✐s r❡✈❡rs❡❞

✷❊● ✫ P❡r❡ts✳✱ ✷✵✶✺ ✭✐♥ ♣r❡♣✳✮

❇✐♥❛r② ♣❧❛♥❡t❡s✐♠❛❧s ♦❢ ♠❛ss ♠ ∼ ✶✵✷✸❣ ♠❡r❣❡ ✇✐t❤✐♥ ❞✐s❦ ❧✐❢❡t✐♠❡s

0.2 0.4 0.6 0.8 0.05 0.1 0.15 0.2

Time [Myr] Binary separation [Rhill]

ep=0, ebin=0 ep=0, ebin=0.5 ep=0.1, ebin=0 ep=0.3, ebin=0 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5

Time [Myr] ebin

0.2 0.4 0.6 0.8 0.05 0.1 0.15 0.2 0.25 0.3

Time [Myr] Orbital e

0.2 0.4 0.6 0.8 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5

Time [Myrs] log (Mach)

■♠♣❧✐❝❛t✐♦♥s ♦♥ P❧❛♥❡t ❢♦r♠❛t✐♦♥ t❤❡♦r②

P❧❛♥❡t❡s✐♠❛❧ ❞✐s❦ ❡✈♦❧✉t✐♦♥ ✭●♦❧❞r❡✐❝❤ ❡t✳ ❛❧✳✱ ✷✵✵✹✮ ❆❞❞✐t✐♦♥❛❧ ❝♦♦❧✐♥❣ t❡r♠

◆❛t✉r❛❧ ♠❡❝❤❛♥✐s♠ ❢♦r ❡❝❝❡♥tr✐❝✐t② ❞❛♠♣✐♥❣

❈♦♥s✐❞❡r❛❜❧❡ ♠❡r❣❡r r❛t❡ ♦❢ ❇Ps

❡♥❤❛♥❝❡ t❤❡ r❛t❡ ♦❢ ❜✐♥❛r② ❤❛r❞❡♥✐♥❣✱ ❝❛t❛❧✐③❡ ❡♥❝♦✉♥t❡r r❛t❡ ❛❞❞✐t✐♦♥❛❧ ❤❡❛t s♦✉r❝❡ ♦❢ t❤❡ ♣❧❛♥❡t❡s✐♠❛❧ ❞✐s❦

❙✉♣❡r✲❊❛rt❤ ❢♦r♠❛t✐♦♥ ✭❍❛♥s❡♥ ✫ ▼✉rr❛②✳✱ ✷✵✶✷✮ ■♥ s✐t✉ ❢♦r♠❛t✐♦♥ ♦❢ s✉♣❡r ❊❛rt❤s ✐s ❝❤❛❧❧❡♥❣✐♥❣

♣♦ss✐❜❧❡ ✐❢ ✐♥✐t✐❛❧ r♦❝❦② ♠❛t❡r✐❛❧ ❡♥❤❛♥❝❡❞ ❜② ❢❛❝t♦r ♦❢ ✷✵

❘❛❞✐❛❧ ❞r✐❢t ❜② ●❉❋ ✐s ❛ ♥❛t✉r❛❧ s♦✉r❝❡ ♣r❡♣❧❛♥❡t❛r② r♦❝❦② ♠❛t❡r✐❛❧

❙✉♠♠❛r②

❖❜s❡r✈❛t✐♦♥❛❧ ❝♦♥str❛✐♥s ✐♠♣❧② ❢❛st ❣r♦✇t❤ ❛♥❞ ❝♦♥s✐❞❡r❛❜❧❡ ♠✐❣r❛t✐♦♥ ❉✐✛❡r❡♥t ♠❛ss r❛♥❣❡s ❛r❡ ❞♦♠✐♥❛t❡❞ ❜② ❞✐✛❡r❡♥t ❣❛s ♣❧❛♥❡t❡s✐♠❛❧ ✐♥t❡r❛❝t✐♦♥s

• ❉❋ ✐s ✐♠♣♦rt❛♥t ❢♦r ❞②♥❛♠✐❝❛❧ ❡✈♦❧✉t✐♦♥ ♦❢ ✐♥t❡r♠❡❞✐❛t❡

♠❛ss ♣❧❛♥❡t❡s✐♠❛❧s

• ❉❋ ❦❡❡♣s ♣❧❛♥❡t❡s✐♠❛❧ ❞✐s❦s ❝♦♦❧ ✇✐t❤ ❧♦✇ r❛♥❞♦♠ ✈❡❧♦❝✐t②
• ❉❋ ❛ss✐sts ✐♥ ♠❡r❣✐♥❣ ❇Ps✱ ✐♥❝r❡❛s❡s ❜✐♥❛r② ❤❛r❞❡♥✐♥❣ r❛t❡

❛♥❞ ❛❞❞s ❤❡❛t♦ t♦ t❤❡ ❞✐s❦

• ❉❋ ♠❛② ❛ss✐st ✐♥ ❜r✐❞❣✐♥❣ ❜❡t✇❡❡♥ ♣❧❛♥❡t ❢♦r♠❛t✐♦♥ t❤❡♦r②

❛♥❞ ❡①♦♣❧❛♥❡t ♦❜s❡r✈❛t✐♦♥s

• ❉❋ ❞♦♠✐♥❛t❡s ♦❢ t②♣❡ ■ ♠✐❣r❛t✐♦♥ ❢♦r ♠♦st r❛♥❣❡ ♦❢ ▼❛❝❤

♥✉♠❜❡rs

▼✐❣r❛t✐♦♥ t♦rq✉❡ ✭❚❛♥❛❦❛ ❡❧✳ ❛❧✱ ✷✵✵✷✮ ❚■ ∼ Σ❣Ω✷❛✹ (▼♣/▼⋆)✷ ❍−✷

✵

s❝❛❧❡s ❛s ▼✷

♣, ✐♥❞❡♣❡♥❞❡♥t ♦❢ M

• ❉❋ ❢♦r♠✉❧❛ ❛♣♣❧✐❝❛❜❧❡ ♦♥❧② ❢♦r

▼♣ ✶✵✷✻❣ ▲✐♠✐t❛t✐♦♥s✿

♥♦♥✲❧✐♥❡❛r r❡❣✐♠❡ ❛❝❝r❡t✐♦♥ s❤❡❛r

❏✸ ✲ ✸❉ ●❉❋ ✭❖❙tr✐❦❡r ✶✾✾✾✮ ❏✷ ✲ ✷❉ ✈❡rt✐❝❛❧❧② ❛✈❡r❛❣❡❞

• ❉❋ ✭▼✉t♦ ❡t✳ ❛❧✱ ✷✵✶✶✮

❉②♥❛♠✐❝❛❧ ❋r✐❝t✐♦♥ ✐♥ ●❛s❡♦✉s ▼❡❞✐✉♠ ✭●❉❋✮

• ♦✈❡r♥✐♥❣ ❡q✉❛t✐♦♥s ✭❖str✐❦❡r✱ ✶✾✾✾✮

❈♦♥t✐♥✉✐t② ❡q✉❛t✐♦♥✿ ∂tρ +∇·(ρ✈) = ✵ ▼♦♠❡♥t✉♠ ❡q✉❛t✐♦♥✿ ∂t✈ +(✈ ·∇)✈ = −∇♣/ρ −∇Φ❡①t ❆♣♣❧②✐♥❣ ❧✐♥❡❛r ♣❡rt✉r❜❛t✐♦♥ ❛♥②❧✐s✐s ②✐❡❧❞s ✐♥❤♦♠♦❣❡♥✉♦✉s ✇❛✈❡ ❡q✉❛t✐♦♥✿ ∇✷α(①,t)− ✶ ❝✷

s

∂ttα(①,t) = −∇✷Φ❡①t(①,t)/❝✷

s

❚❤❡ ❞❡♥s✐t② ✇❛❦❡ ♣r♦♣♦❣❛t❡ ❛s ❛ ♣r❡ss✉r❡ ✇❛✈❡ ✇✐t❤ s♣❡❡❞ ❝s

❖r✐❣✐♥ ♦❢ ❘❡t❛t✐✈❡ ❱❡❧♦❝✐t② ❜❡t✇❡❡♠ ❣❛s ❛♥❞ ♣❧❛♥❡t❡s✐♠❛❧s

❚❤❡ ❣❛s❡♦✉s ❞✐s❦ ✐s s✉❜✲❑❡♣❧❡r✐❛♥ ❞✉❡ t♦ ♣r❡ss✉r❡ ❣r❛❞✐❡♥ts ✈✷

ϕ,❣❛s = ●▼⋆/r + r ρ ❞P ❞r

❙❡tt✐♥❣ P ∼ r−α ✇❡ ❣❡t ✈ϕ,❣❛s = ✈❑(✶−✸·❍✷

✵)✶/✷

❚❤❡ r❡❧❛t✐✈❡ ✈❡❧♦❝✐t② ♦❢ ❛ ♣❧❛♥❡t❡s✐♠❛❧ ✐♥ ❝✐r❝✉❧❛r ♦r❜✐t ✈r❡❧/✈❑ = |✈❑ −✈ϕ,❣❛s|/✈❑ ∼ ❍✷

✵

❚❤❡ ✢♦✇ ✐s s✉❜s♦♥✐❝ ✈r❡❧ ≪ ❝s ❊❝❝❡♥tr✐❝ ❛♥❞ ✐♥❝❧✐♥❡❞ ♦r❜✐ts ✇✐t❤ ❡,■ ❍✵ ❛r❡ s✉♣❡rs♦♥✐❝ ✲ ✈r❡❧ ❝s

❚✉❜✉❧❡♥❝❡

❑♦❧♠♦❣♦r♦✈ ✲ t❤❡ ✢♦✇ ❝♦♥s✐sts ♦❢ s❡❧❢✲s✐♠✐❧❛r ❡❞❞✐❡s ❊♥❡r❣② ❝❛s❝❛❞❡s ❢r♦♠ t❤❡ ❧❛r❣❡st ❡❞❞② t♦ t❤❡ s♠❛❧❧❡st ♦♥❡ ❚②♣✐❝❛❧ ❞✐♠❡♥t✐♦♥s ❧✵ ∼ ❤✱ ✈✵ ∼ ❝s✱ t✵ ∼ ❧✵/✈✵ ∼ ✶/Ω ❆❢t❡r ❛ ❢❡✇ t✵✱ t❤❡ ❣❛s ✐s ✇❡❧❧ ♠✐①❡❞ ✲ s❝❛❧❡❞ ♦❢ ❝st ∼ ❤ ❛r❡ ❞❡str♦②❡❞ ❙♠❛❧❧ s❝❛❧❡s ❛r❡ ✐♥t❛❝t tη ∼ (❧/❧✵)✷/✸t✵ ❛♥❞ t❧/t = (Ωt)−✶/✸✳ ❋♦r t ≪ ✶/Ω✱ t❤❡ ♣❡rt✉r❜❛t✐♦♥ ✐s ♥♦t ❛✛❡❝t❡❞ ❜② t❤❡ ❡❞❞② ❝✉rr❡♥t✱ ❋♦r t ∼ ✶/Ω t❤❡ t✉r❜✉❧❡♥t ❝✉rr❡♥t ♦❢ t❤❡ ❧❛r❣❡st ❡❞❞② ❞❡str♦②s t❤❡ ✇❛❦❡