def L3_integrate(
num_prof,
R,
M,
P_s,
T_s,
rho_s,
R1,
R2,
mat_id_L1,
T_rho_type_id_L1,
T_rho_args_L1,
mat_id_L2,
T_rho_type_id_L2,
T_rho_args_L2,
mat_id_L3,
T_rho_type_id_L3,
T_rho_args_L3,
):
"""Integration of a 2 layer spherical planet.
Parameters
----------
num_prof : int
Number of profile integration steps.
R : float
Radii of the planet (m).
M : float
Mass of the planet (kg).
P_s : float
Pressure at the surface (Pa).
T_s : float
Temperature at the surface (K).
rho_s : float
Density at the surface (kg m^-3).
R1 : float
Boundary between layers 1 and 2 (m).
R2 : float
Boundary between layers 2 and 3 (m).
mat_id_L1 : int
Material id for layer 1.
T_rho_type_id_L1 : int
Relation between A1_T and A1_rho to be used in layer 1.
T_rho_args_L1 : [float]
Extra arguments to determine the relation in layer 1.
mat_id_L2 : int
Material id for layer 2.
T_rho_type_id_L2 : int
Relation between A1_T and A1_rho to be used in layer 2.
T_rho_args_L2 : [float]
Extra arguments to determine the relation in layer 2.
mat_id_L3 : int
Material id for layer 3.
T_rho_type_id_L3 : int
Relation between A1_T and A1_rho to be used in layer 3.
T_rho_args_L3 : [float]
Extra arguments to determine the relation in layer 3.
Returns
-------
A1_r : [float]
The profile radii, in increasing order (m).
A1_m_enc : [float]
The cummulative mass at each profile radius (kg).
A1_P : [float]
The pressure at each profile radius (Pa).
A1_T : [float]
The temperature at each profile radius (K).
A1_rho : [float]
The density at each profile radius (kg m^-3).
A1_u : [float]
The specific internal energy at each profile radius (J kg^-1).
A1_mat_id : [float]
The ID of the material at each profile radius.
"""
A1_r = np.linspace(R, 0, int(num_prof))
A1_m_enc = np.zeros(A1_r.shape)
A1_P = np.zeros(A1_r.shape)
A1_T = np.zeros(A1_r.shape)
A1_rho = np.zeros(A1_r.shape)
A1_u = np.zeros(A1_r.shape)
A1_mat_id = np.zeros(A1_r.shape)
u_s = eos.u_rho_T(rho_s, T_s, mat_id_L3)
T_rho_args_L3 = set_T_rho_args(T_s, rho_s, T_rho_type_id_L3, T_rho_args_L3,
mat_id_L3)
dr = A1_r[0] - A1_r[1]
A1_m_enc[0] = M
A1_P[0] = P_s
A1_T[0] = T_s
A1_rho[0] = rho_s
A1_u[0] = u_s
A1_mat_id[0] = mat_id_L3
for i in range(1, A1_r.shape[0]):
# Layer 3
if A1_r[i] > R2:
rho = A1_rho[i - 1]
mat_id = mat_id_L3
T_rho_type_id = T_rho_type_id_L3
T_rho_args = T_rho_args_L3
rho0 = rho
# Layer 2, 3 boundary
elif A1_r[i] <= R2 and A1_r[i - 1] > R2:
# New density, continuous temperature unless fixed entropy
if T_rho_type_id_L2 == gv.type_ent:
rho = eos.find_rho(
A1_P[i - 1],
mat_id_L2,
T_rho_type_id_L2,
T_rho_args_L2,
A1_rho[i - 1],
1e5,
)
else:
rho = eos.rho_P_T(A1_P[i - 1], A1_T[i - 1], mat_id_L2)
T_rho_args_L2 = set_T_rho_args(A1_T[i - 1], rho,
T_rho_type_id_L2, T_rho_args_L2,
mat_id_L2)
mat_id = mat_id_L2
T_rho_type_id = T_rho_type_id_L2
T_rho_args = T_rho_args_L2
rho0 = A1_rho[i - 1]
# Layer 2
elif A1_r[i] > R1:
rho = A1_rho[i - 1]
mat_id = mat_id_L2
T_rho_type_id = T_rho_type_id_L2
T_rho_args = T_rho_args_L2
rho0 = rho
# Layer 1, 2 boundary
elif A1_r[i] <= R1 and A1_r[i - 1] > R1:
# New density, continuous temperature unless fixed entropy
if T_rho_type_id_L1 == gv.type_ent:
rho = eos.find_rho(
A1_P[i - 1],
mat_id_L1,
T_rho_type_id_L1,
T_rho_args_L1,
A1_rho[i - 1],
1e5,
)
else:
rho = eos.rho_P_T(A1_P[i - 1], A1_T[i - 1], mat_id_L1)
T_rho_args_L1 = set_T_rho_args(A1_T[i - 1], rho,
T_rho_type_id_L1, T_rho_args_L1,
mat_id_L1)
mat_id = mat_id_L1
T_rho_type_id = T_rho_type_id_L1
T_rho_args = T_rho_args_L1
rho0 = A1_rho[i - 1]
# Layer 1
elif A1_r[i] <= R1:
rho = A1_rho[i - 1]
mat_id = mat_id_L1
T_rho_type_id = T_rho_type_id_L1
T_rho_args = T_rho_args_L1
rho0 = A1_rho[i - 1]
A1_m_enc[i] = A1_m_enc[i - 1] - 4 * np.pi * A1_r[i - 1]**2 * rho * dr
A1_P[i] = A1_P[i - 1] + gv.G * A1_m_enc[i - 1] * rho / (A1_r[i - 1]**
2) * dr
A1_rho[i] = eos.find_rho(A1_P[i], mat_id, T_rho_type_id, T_rho_args,
rho0, 1.1 * rho)
A1_T[i] = T_rho(A1_rho[i], T_rho_type_id, T_rho_args, mat_id)
A1_u[i] = eos.u_rho_T(A1_rho[i], A1_T[i], mat_id)
A1_mat_id[i] = mat_id
# Update the T-rho parameters
if mat_id == gv.id_HM80_HHe and T_rho_type_id == gv.type_adb:
T_rho_args = set_T_rho_args(A1_T[i], A1_rho[i], T_rho_type_id,
T_rho_args, mat_id)
if A1_m_enc[i] < 0:
break
return A1_r, A1_m_enc, A1_P, A1_T, A1_rho, A1_u, A1_mat_id
def L1_integrate_out(
r, dr, m_enc, P, T, u, mat_id, T_rho_type_id, T_rho_args, rho_min=0, P_min=0
):
"""Integrate a new layer of a spherical planet outwards.
Parameters
----------
r : float
The radius at the base (m).
dr : float
The radius step for the integration (m).
m_enc : float
The enclosed mass at the base (Pa).
P : float
The pressure at the base (Pa).
T : float
The temperature at the base (K).
u : float
The specific internal energy at the base (J kg^-1).
mat_id : int
The material ID.
T_rho_type_id : int
The ID for the temperature-density relation.
T_rho_args : [float]
Extra arguments for the temperature-density relation.
rho_min : float
The minimum density (must be >= 0) at which the new layer will stop.
P_min : float
The minimum pressure (must be >= 0) at which the new layer will stop.
Returns
-------
A1_r : [float]
The profile radii, in increasing order (m).
A1_m_enc : [float]
The cummulative mass at each profile radius (kg).
A1_P : [float]
The pressure at each profile radius (Pa).
A1_T : [float]
The temperature at each profile radius (K).
A1_rho : [float]
The density at each profile radius (kg m^-3).
A1_u : [float]
The specific internal energy at each profile radius (J kg^-1).
A1_mat_id : [float]
The ID of the material at each profile radius.
"""
# Initialise the profile arrays
A1_r = [r]
A1_m_enc = [m_enc]
A1_P = [P]
A1_T = [T]
A1_u = [u]
A1_mat_id = [mat_id]
A1_rho = [eos.rho_P_T(A1_P[0], A1_T[0], mat_id)]
# Integrate outwards until the minimum density (or zero pressure)
while A1_rho[-1] > rho_min and A1_P[-1] > P_min:
A1_r.append(A1_r[-1] + dr)
A1_m_enc.append(
A1_m_enc[-1] + 4 * np.pi * A1_r[-1] * A1_r[-1] * A1_rho[-1] * dr
)
A1_P.append(A1_P[-1] - gv.G * A1_m_enc[-1] * A1_rho[-1] / (A1_r[-1] ** 2) * dr)
if A1_P[-1] <= 0:
# Add dummy values which will be removed along with the -ve P
A1_rho.append(0)
A1_T.append(0)
A1_u.append(0)
A1_mat_id.append(0)
break
# Update the T-rho parameters
if T_rho_type_id == gv.type_adb and mat_id == gv.id_HM80_HHe:
T_rho_args = set_T_rho_args(
A1_T[-1],
A1_rho[-1],
T_rho_type_id,
T_rho_args,
mat_id,
)
rho = eos.find_rho(
A1_P[-1],
mat_id,
T_rho_type_id,
T_rho_args,
0.9 * A1_rho[-1],
A1_rho[-1],
)
A1_rho.append(rho)
A1_T.append(T_rho(rho, T_rho_type_id, T_rho_args, mat_id))
A1_u.append(eos.u_rho_T(rho, A1_T[-1], mat_id))
A1_mat_id.append(mat_id)
# Remove the duplicate first step and the final too-low density or too-low
# pressure step
return (
A1_r[1:-1],
A1_m_enc[1:-1],
A1_P[1:-1],
A1_T[1:-1],
A1_rho[1:-1],
A1_u[1:-1],
A1_mat_id[1:-1],
)
def L1_integrate(num_prof, R, M, P_s, T_s, rho_s, mat_id, T_rho_type_id, T_rho_args):
"""Integration of a 1 layer spherical planet.
Parameters
----------
num_prof : int
Number of profile integration steps.
R : float
Radii of the planet (m).
M : float
Mass of the planet (kg).
P_s : float
Pressure at the surface (Pa).
T_s : float
Temperature at the surface (K).
rho_s : float
Density at the surface (kg m^-3).
mat_id : int
Material id.
T_rho_type_id : int
Relation between A1_T and A1_rho to be used.
T_rho_args : [float]
Extra arguments to determine the relation.
Returns
-------
A1_r : [float]
The profile radii, in increasing order (m).
A1_m_enc : [float]
The cummulative mass at each profile radius (kg).
A1_P : [float]
The pressure at each profile radius (Pa).
A1_T : [float]
The temperature at each profile radius (K).
A1_rho : [float]
The density at each profile radius (kg m^-3).
A1_u : [float]
The specific internal energy at each profile radius (J kg^-1).
A1_mat_id : [float]
The ID of the material at each profile radius.
"""
# Initialise the profile arrays
A1_r = np.linspace(R, 0, int(num_prof))
A1_m_enc = np.zeros(A1_r.shape)
A1_P = np.zeros(A1_r.shape)
A1_T = np.zeros(A1_r.shape)
A1_rho = np.zeros(A1_r.shape)
A1_u = np.zeros(A1_r.shape)
A1_mat_id = np.ones(A1_r.shape) * mat_id
u_s = eos.u_rho_T(rho_s, T_s, mat_id)
# Set the T-rho relation parameters
T_rho_args = set_T_rho_args(T_s, rho_s, T_rho_type_id, T_rho_args, mat_id)
dr = A1_r[0] - A1_r[1]
# Set the surface values
A1_m_enc[0] = M
A1_P[0] = P_s
A1_T[0] = T_s
A1_rho[0] = rho_s
A1_u[0] = u_s
# Integrate inwards
for i in range(1, A1_r.shape[0]):
A1_m_enc[i] = (
A1_m_enc[i - 1] - 4 * np.pi * A1_r[i - 1] ** 2 * A1_rho[i - 1] * dr
)
A1_P[i] = (
A1_P[i - 1]
+ gv.G * A1_m_enc[i - 1] * A1_rho[i - 1] / (A1_r[i - 1] ** 2) * dr
)
A1_rho[i] = eos.find_rho(
A1_P[i],
mat_id,
T_rho_type_id,
T_rho_args,
A1_rho[i - 1],
1.1 * A1_rho[i - 1],
)
A1_T[i] = T_rho(A1_rho[i], T_rho_type_id, T_rho_args, mat_id)
A1_u[i] = eos.u_rho_T(A1_rho[i], A1_T[i], mat_id)
# Update the T-rho parameters
if mat_id == gv.id_HM80_HHe and T_rho_type_id == gv.type_adb:
T_rho_args = set_T_rho_args(
A1_T[i], A1_rho[i], T_rho_type_id, T_rho_args, mat_id
)
# Stop if run out of mass
if A1_m_enc[i] < 0:
return A1_r, A1_m_enc, A1_P, A1_T, A1_rho, A1_u, A1_mat_id
return A1_r, A1_m_enc, A1_P, A1_T, A1_rho, A1_u, A1_mat_id
def L1_rho_eq_po_from_V(
A1_r_eq,
A1_V_eq,
A1_r_po,
A1_V_po,
P_0,
P_s,
rho_0,
rho_s,
mat_id_L1,
T_rho_type_id_L1,
T_rho_args_L1,
):
"""Compute densities of equatorial and polar profiles given the potential
for a 1 layer planet.
Parameters
----------
A1_r_eq : [float]
Points at equatorial profile where the solution is defined (m).
A1_V_eq : [float]
Equatorial profile of potential (J).
A1_r_po : [float]
Points at equatorial profile where the solution is defined (m).
A1_V_po : [float]
Polar profile of potential (J).
P_0 : float
Pressure at the center of the planet (Pa).
P_s : float
Pressure at the surface of the planet (Pa).
rho_0 : float
Density at the center of the planet (kg m^-3).
rho_s : float
Density at the surface of the planet (kg m^-3).
mat_id_L1 : int
Material id for layer 1.
T_rho_type_id_L1 : int
Relation between T and rho to be used in layer 1.
T_rho_args_L1 : [float]
Extra arguments to determine the relation in layer 1.
Returns
-------
A1_rho_eq : [float]
Equatorial profile of densities (kg m^-3).
A1_rho_po : [float]
Polar profile of densities (kg m^-3).
"""
A1_P_eq = np.zeros(A1_V_eq.shape[0])
A1_P_po = np.zeros(A1_V_po.shape[0])
A1_rho_eq = np.zeros(A1_V_eq.shape[0])
A1_rho_po = np.zeros(A1_V_po.shape[0])
A1_P_eq[0] = P_0
A1_P_po[0] = P_0
A1_rho_eq[0] = rho_0
A1_rho_po[0] = rho_0
# equatorial profile
for i in range(A1_r_eq.shape[0] - 1):
gradV = A1_V_eq[i + 1] - A1_V_eq[i]
gradP = -A1_rho_eq[i] * gradV
A1_P_eq[i + 1] = A1_P_eq[i] + gradP
# avoid overspin
if A1_P_eq[i + 1] > A1_P_eq[i]:
A1_rho_eq[i + 1:] = A1_rho_eq[i]
break
# compute density
if A1_P_eq[i + 1] >= P_s:
A1_rho_eq[i + 1] = eos.find_rho(
A1_P_eq[i + 1],
mat_id_L1,
T_rho_type_id_L1,
T_rho_args_L1,
rho_s * 0.1,
A1_rho_eq[i],
)
else:
A1_rho_eq[i + 1] = 0.0
break
# polar profile
for i in range(A1_r_po.shape[0] - 1):
gradV = A1_V_po[i + 1] - A1_V_po[i]
gradP = -A1_rho_po[i] * gradV
A1_P_po[i + 1] = A1_P_po[i] + gradP
# avoid overspin
if A1_P_po[i + 1] > A1_P_po[i]:
A1_rho_po[i + 1:] = A1_rho_po[i]
break
# compute density
if A1_P_po[i + 1] >= P_s:
A1_rho_po[i + 1] = eos.find_rho(
A1_P_po[i + 1],
mat_id_L1,
T_rho_type_id_L1,
T_rho_args_L1,
rho_s * 0.1,
A1_rho_po[i],
)
else:
A1_rho_po[i + 1] = 0.0
break
return A1_rho_eq, A1_rho_po