diff --git a/src/fragmentation/fragmentation.f90 b/src/fragmentation/fragmentation.f90 index d809bc03a..a01af206b 100644 --- a/src/fragmentation/fragmentation.f90 +++ b/src/fragmentation/fragmentation.f90 @@ -19,14 +19,261 @@ module subroutine fragmentation_regime(Mcb, m1, m2, rad1, rad2, xh1, xh2, vb1, v !! Outcomes during the End Stage of Planet Formation. ApJ 751, 32. https://doi.org/10.1088/0004-637X/751/1/32 !! implicit none - ! Arguments + ! Arguments integer(I4B), intent(out) :: regime real(DP), intent(out) :: Mlr, Mslr real(DP), intent(in) :: Mcb, m1, m2, rad1, rad2, den1, den2, mtiny real(DP), dimension(:), intent(in) :: xh1, xh2, vb1, vb2 real(DP), intent(out) :: Qloss !! The residual energy after the collision + ! Constants + integer(I4B), parameter :: N1 = 1 !number of objects with mass equal to the largest remnant from LS12 + integer(I4B), parameter :: N2 = 2 !number of objects with mass larger than second largest remnant from LS12 + real(DP), parameter :: DENSITY1 = 1000.0_DP !standard density parameter from LS12 [kg/m3] + real(DP), parameter :: MU_BAR = 0.37_DP !0.385#0.37#0.3333# 3.978 # 1/3 material parameter for hydrodynamic planet-size bodies (LS12) + real(DP), parameter :: BETA = 2.85_DP !slope of sfd for remnants from LS12 2.85 + real(DP), parameter :: C1 = 2.43_DP !! Kokubo & Genda (2010) eq. (3) + real(DP), parameter :: C2 = -0.0408_DP !! Kokubo & Genda (2010) eq. (3) + real(DP), parameter :: C3 = 1.86_DP !! Kokubo & Genda (2010) eq. (3) + real(DP), parameter :: C4 = 1.08_DP !! Kokubo & Genda (2010) eq. (3) + real(DP), parameter :: CRUFU = 2.0_DP - 3 * MU_BAR ! central potential variable from Rufu and Aharonson (2019) + real(DP), parameter :: SUPERCAT_QRATIO = 1.8_DP ! See Section 4.1 of LS12 + ! Internals + real(DP) :: a1, alpha, aint, b, bcrit, c_star, egy, zeta, l, lint, mu, phi, theta + real(DP) :: Qr, Qrd_pstar, Qr_erosion, Qr_supercat + real(DP) :: Vhr, Verosion, Vescp, Vhill, Vimp, Vsupercat + real(DP) :: Mint, Mtot + real(DP) :: Rp, rhill + real(DP) :: Mresidual + real(DP) :: U_binding + + Vimp = norm2(vb2(:) - vb1(:)) + b = calc_b(xh2, vb2, xh1, vb1) + l = (rad1 + rad2) * (1 - b) + egy = 0.5_DP * dot_product(vb1, vb1) - GC * Mcb / norm2(xh1) + a1 = - GC * Mcb / 2.0_DP / egy + Mtot = m1 + m2 + mu = (m1 * m2) / Mtot + if (l < 2 * rad2) then + !calculate Mint + phi = 2 * acos((l - rad2) / rad2) + aint = rad2**2 * (PI - (phi - sin(phi)) / 2.0_DP) + lint = 2 * sqrt(rad2**2 - (rad2 - l / 2.0_DP) ** 2) + Mint = aint * lint ![kg] + alpha = (l**2) * (3 * rad2 - l) / (4 * (rad2**3)) + else + alpha = 1.0_DP + Mint = m2 + end if + Rp = (3 * (m1 / den1 + alpha * m2 / den2) / (4 * PI))**(1.0_DP/3.0_DP) ! (Mustill et al. 2018) + c_star = calc_c_star(Rp) + !calculate Vescp + Vescp = sqrt(2 * GC * Mtot / Rp) !Mustill et al. 2018 eq 6 + !calculate rhill + rhill = a1 * (m1 / 3.0_DP / (Mcb + m1))**(1.0_DP/3.0_DP) + !calculate Vhill + if ((rad2 + rad1) < rhill) then + Vhill = sqrt(2 * GC * m1 * ((rhill**2 - rhill * (rad1 + rad2)) / & + (rhill**2 - 0.5_DP * (rad1 + rad2)**2)) / (rad1 + rad2)) + else + Vhill = Vescp + end if + !calculate Qr_pstar + Qrd_pstar = calc_Qrd_pstar(m1, m2, alpha, c_star) * (Vhill / Vescp)**CRUFU !Rufu and Aharaonson eq (3) + !calculate Verosion + Qr_erosion = 2 * (1.0_DP - m1 / Mtot) * Qrd_pstar + Verosion = (2 * Qr_erosion * Mtot / mu)** (1.0_DP / 2.0_DP) + Qr = mu*(Vimp**2) / Mtot / 2.0_DP + !calculate mass largest remnant Mlr + Mlr = (1.0_DP - Qr / Qrd_pstar / 2.0_DP) * Mtot ! [kg] # LS12 eq (5) + !calculate Vsupercat + Qr_supercat = SUPERCAT_QRATIO * Qrd_pstar ! See LS12 Section 4.1 + Vsupercat = sqrt(2 * Qr_supercat * Mtot / mu) + !calculate Vhr + zeta = (m1 - m2) / Mtot + theta = 1.0_DP - b + Vhr = Vescp * (C1 * zeta**2 * theta**(2.5_DP) + C2 * zeta**2 + C3 * theta**(2.5_DP) + C4) ! Kokubo & Genda (2010) eq. (3) + bcrit = rad1 / (rad1 + rad2) + Qloss = 0.0_DP + U_binding = (3.0_DP * Mtot) / (5.0_DP * Rp) ! LS12 eq. 27 + + if ((m1 < mtiny).or.(m2 < mtiny)) then + regime = COLLRESOLVE_REGIME_MERGE !perfect merging regime + Mlr = Mtot + Mslr = 0.0_DP + Qloss = 0.0_DP + write(*,*) "FORCE MERGE" + else + if( Vimp < Vescp) then + regime = COLLRESOLVE_REGIME_MERGE !perfect merging regime + Mlr = Mtot + Mslr = 0.0_DP + Qloss = 0.0_DP + else if (Vimp < Verosion) then + if (b < bcrit) then + regime = COLLRESOLVE_REGIME_MERGE !partial accretion regime" + Mlr = Mtot + Mslr = 0.0_DP + Qloss = 0.0_DP + else if ((b > bcrit) .and. (Vimp < Vhr)) then + regime = COLLRESOLVE_REGIME_MERGE ! graze and merge + Mlr = Mtot + Mslr = 0.0_DP + Qloss = 0.0_DP + else + Mlr = m1 + Mslr = calc_Qrd_rev(m2, m1, Mint, den1, den2, Vimp, c_star) + regime = COLLRESOLVE_REGIME_HIT_AND_RUN !hit and run + Qloss = (c_star + 1.0_DP) * U_binding ! Qr + end if + else if (Vimp > Verosion .and. Vimp < Vsupercat) then + if (m2 < 0.001_DP * m1) then + regime = COLLRESOLVE_REGIME_MERGE !cratering regime" + Mlr = Mtot + Mslr = 0.0_DP + Qloss = 0.0_DP + else + Mslr = Mtot * (3.0_DP - BETA) * (1.0_DP - N1 * Mlr / Mtot) / (N2 * BETA) ! LS12 eq (37) + regime = COLLRESOLVE_REGIME_DISRUPTION !disruption + Qloss = (c_star + 1.0_DP) * U_binding ! Qr - Qr_erosion + end if + else if (Vimp > Vsupercat) then + Mlr = Mtot * 0.1_DP * (Qr / (Qrd_pstar * SUPERCAT_QRATIO))**(-1.5_DP) !LS12 eq (44) + Mslr = Mtot * (3.0_DP - BETA) * (1.0_DP - N1 * Mlr / Mtot) / (N2 * BETA) !LS12 eq (37) + regime = COLLRESOLVE_REGIME_SUPERCATASTROPHIC ! supercatastrophic + Qloss = (c_star + 1.0_DP) * U_binding ! Qr - Qr_supercat + else + write(*,*) "Error no regime found in symba_regime" + end if + end if + Mresidual = Mtot - Mlr - Mslr + if (Mresidual < 0.0_DP) then ! prevents final masses from going negative + Mlr = Mlr + Mresidual + end if + + return + + ! Internal functions + contains + function calc_Qrd_pstar(Mtarg, Mp, alpha, c_star) result(Qrd_pstar) + !! author: Jennifer L.L. Pouplin and Carlisle A. Wishard + !! + !! Calculates the corrected Q* for oblique impacts. See Eq. (15) of LS12. + !! Reference: + !! Leinhardt, Z.M., Stewart, S.T., 2012. Collisions between Gravity-dominated Bodies. I. Outcome Regimes and Scaling + !! Laws 745, 79. https://doi.org/10.1088/0004-637X/745/1/79 + !! + implicit none + ! Arguments + real(DP),intent(in) :: Mtarg, Mp, alpha, c_star + ! Result + real(DP) :: Qrd_pstar + ! Internals + real(DP) :: Qrd_star1, mu_alpha, mu, Qrd_star + + ! calc mu, mu_alpha + mu = (Mtarg * Mp) / (Mtarg + Mp) ! [kg] + mu_alpha = (Mtarg * alpha * Mp) / (Mtarg + alpha * Mp) ! [kg] + ! calc Qrd_star1 + Qrd_star1 = (c_star * 4 * PI * DENSITY1 * GC * Rp**2) / 5.0_DP + ! calc Qrd_star + Qrd_star = Qrd_star1 * (((Mp / Mtarg + 1.0_DP)**2) / (4 * Mp / Mtarg))**(2.0_DP / (3.0_DP * MU_BAR) - 1.0_DP) !(eq 23) + ! calc Qrd_pstar, v_pstar + Qrd_pstar = ((mu / mu_alpha)**(2.0_DP - 3.0_DP * MU_BAR / 2.0_DP)) * Qrd_star ! (eq 15) + + return + end function calc_Qrd_pstar + + function calc_Qrd_rev(Mp, Mtarg, Mint, den1, den2, Vimp, c_star) result(Mslr) + !! author: Jennifer L.L. Pouplin and Carlisle A. Wishard + !! + !! Calculates mass of second largest fragment. + !! + implicit none + ! Arguments + real(DP),intent(in) :: Mp, Mtarg, Mint, den1, den2, Vimp, c_star + ! Result + real(DP) :: Mslr + ! Internals + real(DP) :: mtot_rev, mu_rev, gamma_rev, Qrd_star1, Qrd_star, mu_alpha_rev + real(DP) :: Qrd_pstar, Rc1, Qr_rev, Qrd_pstar_rev, Qr_supercat_rev + + ! calc Mslr, Rc1, mu, gammalr + mtot_rev = Mint + Mp + Rc1 = (3 * (Mint / den1 + Mp / den2) / (4 * PI))**(1.0_DP/3.0_DP) ! [m] Mustill et al 2018 + mu_rev = (Mint * Mp) / mtot_rev ! [kg] eq 49 LS12 + mu_alpha_rev = (Mtarg * alpha * Mp) / (Mtarg + alpha * Mp) + gamma_rev = Mint / Mp ! eq 50 LS12 + !calc Qr_rev + Qr_rev = mu_rev * (Vimp**2) / (2 * mtot_rev) + ! calc Qrd_star1, v_star1 + Qrd_star1 = (c_star * 4 * PI * mtot_rev * GC ) / Rc1 / 5.0_DP + ! calc Qrd_pstar_rev + Qrd_star = Qrd_star1 * (((gamma_rev + 1.0_DP)**2) / (4 * gamma_rev)) ** (2.0_DP / (3.0_DP * MU_BAR) - 1.0_DP) !(eq 52) + Qrd_pstar = Qrd_star * ((mu_rev / mu_alpha_rev)**(2.0_DP - 3.0_DP * MU_BAR / 2.0_DP)) + Qrd_pstar_rev = Qrd_pstar * (Vhill / Vescp)**CRUFU !Rufu and Aharaonson eq (3) + !calc Qr_supercat_rev + Qr_supercat_rev = 1.8_DP * Qrd_pstar_rev + if (Qr_rev > Qr_supercat_rev ) then + Mslr = mtot_rev * (0.1_DP * ((Qr_rev / (Qrd_pstar_rev * 1.8_DP))**(-1.5_DP))) !eq (44) + else if ( Qr_rev < Qrd_pstar_rev ) then + Mslr = Mp + else + Mslr = (1.0_DP - Qr_rev / Qrd_pstar_rev / 2.0_DP) * (mtot_rev) ! [kg] #(eq 5) + end if + + if ( Mslr > Mp ) Mslr = Mp !check conservation of mass + + return + end function calc_Qrd_rev + + function calc_b(proj_pos, proj_vel, targ_pos, targ_vel) result(sintheta) + !! author: Jennifer L.L. Pouplin, Carlisle A. Wishard, and David A. Minton + !! + !! Calculates the impact factor b = sin(theta), where theta is the angle between the relative velocity + !! and distance vectors of the target and projectile bodies. See Fig. 2 of Leinhardt and Stewart (2012) + !! + implicit none + !! Arguments + real(DP), dimension(:), intent(in) :: proj_pos, proj_vel, targ_pos, targ_vel + !! Result + real(DP) :: sintheta + !! Internals + real(DP), dimension(NDIM) :: imp_vel, distance, x_cross_v + + imp_vel(:) = proj_vel(:) - targ_vel(:) + distance(:) = proj_pos(:) - targ_pos(:) + x_cross_v(:) = distance(:) .cross. imp_vel(:) + sintheta = norm2(x_cross_v(:)) / norm2(distance(:)) / norm2(imp_vel(:)) + return + end function calc_b + + function calc_c_star(Rc1) result(c_star) + !! author: David A. Minton + !! + !! Calculates c_star as a function of impact equivalent radius. It inteRpolates between 5 for ~1 km sized bodies to + !! 1.8 for ~10000 km sized bodies. See LS12 Fig. 4 for details. + !! + implicit none + !! Arguments + real(DP), intent(in) :: Rc1 + !! Result + real(DP) :: c_star + !! Internals + real(DP), parameter :: loR = 1.0e3_DP ! Lower bound of inteRpolation size (m) + real(DP), parameter :: hiR = 1.0e7_DP ! Upper bound of inteRpolation size (m) + real(DP), parameter :: loval = 5.0_DP ! Value of C* at lower bound + real(DP), parameter :: hival = 1.9_DP ! Value of C* at upper bound + + if (Rc1 < loR) then + c_star = loval + else if (Rc1 < hiR) then + c_star = loval + (hival - loval) * log(Rc1 / loR) / log(hiR /loR) + else + c_star = hival + end if + return + end function calc_c_star - return end subroutine fragmentation_regime end submodule s_fragmentation \ No newline at end of file