def main():
# Thomas et al. 2005
mass = 9.395e20
gm = mass * pyshtools.constant.G.value
r0 = 476.2e3
omega = 2 * np.pi / (9.076 * 60 * 60)
volume = 4 * np.pi / 3 * r0**3
lmax = 4
# Model C1
radius = np.ndarray(2)
radius[0] = 0.
radius[1] = r0
rho = np.ndarray(2)
rho[0] = mass / volume
rho[1] = 0
print(rho[0])
hlm, clm, mass_model = HydrostaticShape(radius, rho, omega, gm, r0)
a = hlm[1].expand(lat=0., lon=0., lmax_calc=lmax)
b = hlm[1].expand(lat=0., lon=90., lmax_calc=lmax)
c = hlm[1].expand(lat=90., lon=0., lmax_calc=lmax)
print(a, b, c, a - c)
def main():
gravfile = 'Data/gmm3_120_sha.tab'
topofile = 'Data/MarsTopo719.shape'
densityfile = 'Data/dichotomy_359.sh'
model_name = [
'dwcold', 'dwhot_modif', 'dwhot_stairs', 'DWTh2Ref1', 'DWTh2Ref2',
'eh70cold', '"eh70hot', 'Gudkova'
]
spec = 'Data/Mars-reference-interior-models/model_'
interior_file = [spec + name for name in model_name]
lmax_calc = 90
lmax = lmax_calc * 4
potcoefs, lmaxp, header = pyshtools.shio.shread(gravfile,
header=True,
lmax=lmax)
potential = pyshtools.SHCoeffs.from_array(potcoefs)
potential.r_ref = float(header[0]) * 1.e3
potential.gm = float(header[1]) * 1.e9
potential.mass = potential.gm / float(pyshtools.constant.grav_constant)
print('Gravity file = {:s}'.format(gravfile))
print('Lmax of potential coefficients = {:d}'.format(lmaxp))
print('Reference radius (km) = {:f}'.format(potential.r_ref / 1.e3))
print('GM = {:e}\n'.format(potential.gm))
topo = pyshtools.SHCoeffs.from_file(topofile, lmax=lmax)
topo.r0 = topo.coeffs[0, 0, 0]
print('Topography file = {:s}'.format(topofile))
print('Lmax of topography coefficients = {:d}'.format(topo.lmax))
print('Reference radius (km) = {:f}\n'.format(topo.r0 / 1.e3))
density = pyshtools.SHCoeffs.from_file(densityfile, lmax=lmax)
print('Lmax of density coefficients = {:d}\n'.format(density.lmax))
lat_insight = 4.43
lon_insight = 135.84
filter = 1
half = 50
nmax = 7
lmax_hydro = 15
t0_sigma = 5. # maximum difference between minimum crustal thickness
omega = float(pyshtools.constant.omega_mars)
d_lith = 150.e3
t0 = 1.e3 # minimum crustal thickness
model = 3 # identifier for the interior reference model
# --- read 1D reference interior model ---
print('=== Reading model {:s} ==='.format(model_name[model]))
with open(interior_file[model], 'r') as f:
lines = f.readlines()
print(lines[0].strip())
data = lines[1].split()
if float(data[2]) != 1:
raise RuntimeError('Program not capable of reading polynomial ' +
'files')
num = int(lines[2].split()[0])
crust_index = int(lines[2].split()[3])
mantle_index = int(lines[2].split()[3])
radius = np.zeros(num)
rho = np.zeros(num)
for i in range(0, num):
data = lines[i + 3].split()
radius[i] = float(data[0])
rho[i] = float(data[1])
r0_model = radius[num - 1]
print('Surface radius of model (km) = {:f}'.format(r0_model / 1.e3))
for i in range(0, num):
if radius[i] <= (r0_model - d_lith) and \
radius[i+1] > (r0_model - d_lith):
if radius[i] == (r0_model - d_lith):
i_lith = i
elif (r0_model - d_lith) - radius[i] <= radius[i+1] -\
(r0_model - d_lith):
i_lith = i
else:
i_lith = i + 1
break
n = num - 1
rho[n] = 0. # the density above the surface is zero
rho_mantle = rho[crust_index - 1]
print('Mantle density (kg/m3) = {:f}'.format(rho_mantle))
print('Assumed depth of lithosphere (km) = {:f}'.format(d_lith / 1.e3))
print('Actual depth of lithosphere in discretized model (km) = {:f}'.
format((r0_model - radius[i_lith]) / 1.e3))
# --- Compute gravity contribution from hydrostatic density interfaces ---
if False:
# compute values for a planet that is completely fluid
hlm_fluid, clm_fluid, mass_model = \
HydrostaticShape(radius, rho, omega, potential.gm, potential.r_ref,
finiteamplitude=True)
print('--- Hydrostatic potential coefficients for a fluid planet ---')
print('c20 = {:e}\nc40 = {:e}'.format(clm_fluid.coeffs[0, 2, 0],
clm_fluid.coeffs[0, 4, 0]))
print('--- Hydrostatic relief of surface for a fluid planet ---')
print('h20 = {:e}\nh40 = {:e}'.format(hlm_fluid[n].coeffs[0, 2, 0],
hlm_fluid[n].coeffs[0, 4, 0]))
hlm, clm_hydro, mass_model = \
HydrostaticShapeLith(radius, rho, i_lith, potential, omega,
lmax_hydro, finiteamplitude=False)
print('Total mass of model (kg) = {:e}'.format(mass_model))
print('% of J2 arising from beneath lithosphere = {:f}'.format(
clm_hydro.coeffs[0, 2, 0] / potential.coeffs[0, 2, 0] * 100.))
potential.coeffs[:, :lmax_hydro+1, :lmax_hydro+1] -= \
clm_hydro.coeffs[:, :lmax_hydro+1, :lmax_hydro+1]
# --- Constant density model ---
rho_c = 2900.
print('-- Constant density model --\nrho_c = {:f}'.format(rho_c))
tmin = 1.e9
thickave = 44.e3 # initial guess of average crustal thickness
while abs(tmin - t0) > t0_sigma:
# iterate to fit assumed minimum crustal thickness
moho = pyMoho.pyMoho(potential,
topo,
lmax,
rho_c,
rho_mantle,
thickave,
filter_type=filter,
half=half,
lmax_calc=lmax_calc,
nmax=nmax,
quiet=True)
thick_grid = (topo.pad(lmax) - moho.pad(lmax)).expand(grid='DH2')
print('Average crustal thickness (km) = {:f}'.format(thickave / 1.e3))
print('Crustal thickness at InSight landing sites (km) = {:f}'.format(
(topo.pad(lmax) - moho.pad(lmax)).expand(lat=lat_insight,
lon=lon_insight) / 1.e3))
tmin = thick_grid.data.min()
tmax = thick_grid.data.max()
print('Minimum thickness (km) = {:e}'.format(tmin / 1.e3))
print('Maximum thickness (km) = {:e}'.format(tmax / 1.e3))
thickave += t0 - tmin
thick_grid.plot(show=False, fname='Thick-Mars-1.png')
moho.plot_spectrum(show=False, fname='Moho-spectrum-Mars-1.png')
# --- Model with variable density ---
rho_south = 2900.
rho_north = 2900.
porosity = 0.0
print('-- Variable density model ---\n' +
'rho_south = {:f}\n'.format(rho_south) +
'rho_north = {:f}'.format(rho_north))
density = density * (rho_north - rho_south)
density.coeffs[0, 0, 0] += rho_south
tmin = 1.e9
thickave = 44.e3 # initial guess of average crustal thickness
while abs(tmin - t0) > t0_sigma:
# iterate to fit assumed minimum crustal thickness
moho = pyMoho.pyMohoRho(potential,
topo,
density,
porosity,
lmax,
rho_mantle,
thickave,
filter_type=filter,
half=half,
lmax_calc=lmax_calc,
quiet=True,
nmax=nmax)
thick_grid = (topo.pad(lmax) - moho.pad(lmax)).expand(grid='DH2')
print('Average crustal thickness (km) = {:e}'.format(thickave / 1.e3))
print('Crustal thickness at InSight landing sites (km) = {:e}'.format(
(topo.pad(lmax) - moho.pad(lmax)).expand(lat=lat_insight,
lon=lon_insight) / 1.e3))
tmin = thick_grid.data.min()
tmax = thick_grid.data.max()
print('Minimum thickness (km) = {:e}'.format(tmin / 1.e3))
print('Maximum thickness (km) = {:e}'.format(tmax / 1.e3))
thickave += t0 - tmin
thick_grid.plot(show=False, fname='Thick-Mars-2.png')
moho.plot_spectrum(show=False, fname='Moho-spectrum-Mars-2.png')
def main():
model_name = ['PREM.dat']
spec = 'Data/'
interior_file = [spec + name for name in model_name]
r_ref = float(pyshtools.constant.r0_pot_earth)
gm = float(pyshtools.constant.gm_earth)
mass = float(pyshtools.constant.mass_earth)
omega = float(pyshtools.constant.wgs84_omega)
print('Reference radius (km) = {:f}'.format(r_ref / 1.e3))
print('GM = {:e}'.format(gm))
print('Mass = {:e}'.format(mass))
print('Omega = {:e}'.format(omega))
model = 0
# --- read 1D reference interior model ---
print('=== Reading model {:s} ==='.format(model_name[model]))
with open(interior_file[model], 'r') as f:
lines = f.readlines()
print(lines[0].strip())
data = lines[1].split()
if float(data[2]) != 1:
raise RuntimeError('Program not capable of reading polynomial ' +
'files')
num = int(lines[2].split()[0])
crust_index = int(lines[2].split()[3])
mantle_index = int(lines[2].split()[3])
radius = np.zeros(num)
rho = np.zeros(num)
for i in range(0, num):
data = lines[i + 3].split()
radius[i] = float(data[0])
rho[i] = float(data[1])
r0_model = radius[num - 1]
print('Surface radius of model (km) = {:f}'.format(r0_model / 1.e3))
n = num - 1
rho[n] = 0. # the density above the surface is zero
rho_mantle = rho[crust_index - 1]
print('Mantle density (kg/m3) = {:f}'.format(rho_mantle))
# --- Compute gravity contribution from hydrostatic density interfaces ---
if True:
for i in range(1, 8):
# compute values for a planet that is completely fluid
fn = True
nmax = i
hlm_fluid, clm_fluid, mass_model = \
HydrostaticShape(radius, rho, omega, gm,
r_ref, nmax=nmax,
finiteamplitude=fn, kmax=4)
hlm_surf_unnorm = hlm_fluid[n].convert(normalization='unnorm')
print('--- Hydrostatic relief of surface ---')
print('nmax = {:d}'.format(nmax))
print('h20 = {:e}\n'.format(hlm_fluid[n].coeffs[0, 2, 0]) +
'h40 = {:e}'.format(hlm_fluid[n].coeffs[0, 4, 0]))
print('h20 (unnorm)= {:e}\n'.format(
hlm_surf_unnorm.coeffs[0, 2, 0] / r0_model) +
'h40 (unnorm)= {:e}'.format(hlm_surf_unnorm.coeffs[0, 4, 0] /
r0_model))
print(mass, mass_model)
def main():
d_lith = 150.e3
rho_crust = 2900.
d_sigma = 45.e3
lmax_hydro = 90
gravfile = 'Data/gmm3_120_sha.tab'
topofile = 'Data/MarsTopo719.shape'
densityfile = 'Data/dichotomy_359.sh'
model_name = [
'DWThot', 'DWThotCrust1', 'DWThotCrust1r', 'EH45Tcold',
'EH45TcoldCrust1', 'EH45TcoldCrust1r', 'EH45ThotCrust2',
'EH45ThotCrust2r', 'LFAK', 'SANAK', 'TAYAK', 'DWAK', 'ZG_DW'
]
spec = 'Data/Mars-reference-interior-models/Smrekar/'
interior_file = [spec + name + '.deck' for name in model_name]
potential = pyshtools.SHGravCoeffs.from_file(gravfile, header_units='km')
omega = pyshtools.constant.omega_mars.value
potential.omega = omega
mass_mars = np.float_(potential.mass)
print('Gravity file = {:s}'.format(gravfile))
print('Lmax of potential coefficients = {:d}'.format(potential.lmax))
print('Reference radius (km) = {:f}'.format(potential.r0 / 1.e3))
print('GM = {:e}'.format(potential.gm))
print('Mass = {:e}'.format(potential.mass))
print('Omega = {:e}'.format(potential.omega))
lmax_calc = 90
lmax = 359
model = 10
topo = pyshtools.SHCoeffs.from_file(topofile, lmax=lmax)
topo.r0 = topo.coeffs[0, 0, 0]
r_mars = topo.coeffs[0, 0, 0]
print('Topography file = {:s}'.format(topofile))
print('Lmax of topography coefficients = {:d}'.format(topo.lmax))
print('Reference radius (km) = {:f}\n'.format(topo.r0 / 1.e3))
# --- Make geoid map, expand in spherical harmonics
# --- and then remove degree-0 term, for plotting purposes only.
u0 = potential.gm / potential.r0
geoid = potential.geoid(u0,
r=topo.r0,
order=2,
lmax_calc=lmax_calc,
lmax=lmax)
geoidsh = geoid.geoid.expand()
geoidsh.coeffs[0, 0, 0] = 0.
geoid = geoidsh.expand(grid='DH2')
# --- read 1D reference interior model ---
radius, rho, i_crust, i_core, i_lith = ReadRefModel(interior_file[model],
depth=d_lith)
rho_mantle = rho[i_crust - 1]
rho_core = rho[i_core - 1]
n = len(radius) - 1
r0_model = radius[n]
# --- Compute purely hydrostatic relief of all interfaces ---
print('\n=== Fluid planet ===')
if False:
for i in range(1, n + 1):
hlm_fluid, clm_fluid, mass_model = HydrostaticShape(radius,
rho,
omega,
potential.gm,
potential.r0,
i_clm_hydro=i)
print('i = {:d}, r = {:f}, rho = {:f}, %C20 = {:f}'.format(
i, radius[i], rho[i],
clm_fluid.coeffs[0, 2, 0] / potential.coeffs[0, 2, 0] * 100))
hlm_fluid, clm_fluid, mass_model = \
HydrostaticShape(radius, rho, omega, potential.gm, potential.r0)
print('--- Hydrostatic relief of surface for a fluid planet ---')
print('h20 = {:e}\n'.format(hlm_fluid[n].coeffs[0, 2, 0]) +
'h40 = {:e}'.format(hlm_fluid[n].coeffs[0, 4, 0]))
hydro_surface = hlm_fluid[n].expand()
print('Elevation difference between pole and equator (km)'
' {:e}'.format(hydro_surface.max() / 1.e3 -
hydro_surface.min() / 1.e3))
print('--- Hydrostatic relief of core-mantle boundary for a fluid planet '
'---')
print('h20 = {:e}\n'.format(hlm_fluid[i_core].coeffs[0, 2, 0]) +
'h40 = {:e}'.format(hlm_fluid[i_core].coeffs[0, 4, 0]))
print('Moments of hydrostatic core')
II, AA, BB, CC, mass, RR, vec = InertiaTensor_from_shape(hlm_fluid,
rho,
i_core,
quiet=False)
print('I = ', II)
print('A, B, C = ', AA, BB, CC)
print('A, B, C / (mass_mars r0^2) = ',
(AA, BB, CC) / mass_mars / r_mars**2)
print('mass of core (kg) = ', mass)
print('R core (m) = ', RR)
print('Moments of hydrostatic planet')
II, AA, BB, CC, mass, RR, vec = InertiaTensor_from_shape(hlm_fluid,
rho,
n,
quiet=False)
print('I = ', II)
print('A, B, C = ', AA, BB, CC)
print('A, B, C / (mass_mars r0^2) = ',
(AA, BB, CC) / mass_mars / r_mars**2)
print('mass of planet (kg) = ', mass)
print('R surface (m) = ', RR)
# --- Compute relief along hydrostatic interfaces with a lithosphere ---
print('\n=== Planet with a lithosphere ===')
r_sigma = topo.r0 - d_sigma
hlm, clm_hydro, mass_model = \
HydrostaticShapeLith(radius, rho, i_lith, potential, topo, rho_crust,
r_sigma, omega, lmax_hydro)
print('--- Core shape, with lithosphere ---')
for l in range(0, 5):
for m in range(0, l + 1):
print(
l,
m,
hlm[i_core].coeffs[0, l, m],
hlm[i_core].coeffs[1, l, m],
)
print('Max and min of degree-1 core shape =',
hlm[i_core].expand(lmax=1).max(), hlm[i_core].expand(lmax=1).min())
II, AA, BB, CC, mass, RR, vec = InertiaTensor_from_shape(hlm,
rho,
i_core,
quiet=False)
print('I = ', II)
print('A, B, C = ', AA, BB, CC)
print('A, B, C / (mass_mars r0^2) = ',
(AA, BB, CC) / mass_mars / r_mars**2)
print('mass of core (kg) = ', mass)
print('R core (m) = ', RR)
# --- Calculate relief, with respect to hydrostatic solution ---
# --- at i_lith and i_core
print('\n=== Difference between fluid planet and planet with a '
'lithosphere ===')
diff_ilith = hlm[i_lith] - hlm_fluid[i_lith].pad(lmax=lmax_hydro)
grid_ilith = diff_ilith.expand()
print('Maximum and minimum difference at i_lith (km) =- ',
grid_ilith.max() / 1.e3,
grid_ilith.min() / 1.e3)
diff_icore = hlm[i_core] - hlm_fluid[i_core].pad(lmax=lmax_hydro)
grid_icore = diff_icore.expand()
print('Maximum and minimum difference at i_core (km) =- ',
grid_icore.max() / 1.e3,
grid_icore.min() / 1.e3)
# ---- Write data to files ---
print('\n=== Output gridded data for use with GMT ===')
print('Output grid sizes = ', geoid.nlat, geoid.nlon)
(geoid / 1000).to_file('figs/Mars_geoid.dat')
diff = (geoid - hlm_fluid[n].pad(lmax=lmax).expand(grid='DH2') + radius[n])
(diff / 1000).to_file('figs/Mars_geoid_diff.dat')
diff = hlm[i_lith].pad(lmax=lmax).expand(grid='DH2') - radius[i_lith]
(diff / 1000).to_file('figs/hydro_ilith.dat')
diff = hlm[i_core].pad(lmax=lmax).expand(grid='DH2') - radius[i_core]
(diff / 1000).to_file('figs/hydro_icore.dat')
diff = hlm[i_lith] - hlm_fluid[i_lith].pad(lmax=lmax_hydro)
(diff.expand(grid='DH2', lmax=lmax) /
1000).to_file('figs/hydro_ilith_diff.dat')
diff = hlm[i_core] - hlm_fluid[i_core].pad(lmax=lmax_hydro)
(diff.expand(grid='DH2', lmax=lmax) /
1000).to_file('figs/hydro_icore_diff.dat')
# Sensitivity tests
print('\n=== Sensitivity tests ===')
r_sigma = topo.r0
hlm2, clm_hydro2, mass_model2 = \
HydrostaticShapeLith(radius, rho, i_lith, potential, topo, rho_crust,
r_sigma, omega, lmax_hydro)
print('d_sigma = 0')
print('Minimum and maximum differences at i_lith (m) = ',
(hlm2[i_lith] - hlm[i_lith]).expand().min(),
(hlm2[i_lith] - hlm[i_lith]).expand().max())
print('Minimum and maximum differences at i_core (m) = ',
(hlm2[i_core] - hlm[i_core]).expand().min(),
(hlm2[i_core] - hlm[i_core]).expand().max())
r_sigma = topo.r0 - 100.e3
hlm2, clm_hydro2, mass_model2 = \
HydrostaticShapeLith(radius, rho, i_lith, potential, topo, rho_crust,
r_sigma, omega, lmax_hydro)
print('d_sigma = 100 km')
print('Minimum and maximum differences at i_lith (m) = ',
(hlm2[i_lith] - hlm[i_lith]).expand().min(),
(hlm2[i_lith] - hlm[i_lith]).expand().max())
print('Minimum and maximum differences at i_core (m) = ',
(hlm2[i_core] - hlm[i_core]).expand().min(),
(hlm2[i_core] - hlm[i_core]).expand().max())
r_sigma = topo.r0 - d_sigma
hlm2, clm_hydro2, mass_model2 = \
HydrostaticShapeLith(radius, rho, i_lith, potential, topo, 2500.,
r_sigma, omega, lmax_hydro)
print('rho = 2500 kg / m3')
print('Minimum and maximum differences at i_lith (m) = ',
(hlm2[i_lith] - hlm[i_lith]).expand().min(),
(hlm2[i_lith] - hlm[i_lith]).expand().max())
print('Minimum and maximum differences at i_core (m) = ',
(hlm2[i_core] - hlm[i_core]).expand().min(),
(hlm2[i_core] - hlm[i_core]).expand().max())
r_sigma = topo.r0 - d_sigma
hlm2, clm_hydro2, mass_model2 = \
HydrostaticShapeLith(radius, rho, i_lith, potential, topo, 3300.,
r_sigma, omega, lmax_hydro)
print('rho = 3300 kg / m3')
print('Minimum and maximum differences at i_lith (m) = ',
(hlm2[i_lith] - hlm[i_lith]).expand().min(),
(hlm2[i_lith] - hlm[i_lith]).expand().max())
print('Minimum and maximum differences at i_core (m) = ',
(hlm2[i_core] - hlm[i_core]).expand().min(),
(hlm2[i_core] - hlm[i_core]).expand().max())
def main():
gravfile = 'Data/gmm3_120_sha.tab'
topofile = 'Data/MarsTopo719.shape'
densityfile = 'Data/dichotomy_359.sh'
model_name = ['DWThot', 'DWThotCrust1', 'DWThotCrust1r', 'EH45Tcold',
'EH45TcoldCrust1', 'EH45TcoldCrust1r', 'EH45ThotCrust2',
'EH45ThotCrust2r', 'LFAK', 'SANAK', 'TAYAK', 'DWAK',
'ZG_DW']
spec = 'Data/Mars-reference-interior-models/Smrekar/'
interior_file = [spec + name + '.deck' for name in model_name]
lmax_calc = 90
lmax = lmax_calc * 4
potential = pyshtools.SHGravCoeffs.from_file(gravfile, header_units='km')
print('Gravity file = {:s}'.format(gravfile))
print('Lmax of potential coefficients = {:d}'.format(potential.lmax))
print('Reference radius (km) = {:f}'.format(potential.r0 / 1.e3))
print('GM = {:e}\n'.format(potential.gm))
topo = pyshtools.SHCoeffs.from_file(topofile, lmax=lmax)
topo.r0 = topo.coeffs[0, 0, 0]
print('Topography file = {:s}'.format(topofile))
print('Lmax of topography coefficients = {:d}'.format(topo.lmax))
print('Reference radius (km) = {:f}\n'.format(topo.r0 / 1.e3))
density = pyshtools.SHCoeffs.from_file(densityfile, lmax=lmax)
print('Lmax of density coefficients = {:d}\n'.format(density.lmax))
lat_insight = 4.502384
lon_insight = 135.623447
filter = 1
half = 50
nmax = 7
lmax_hydro = 15
t0_sigma = 5. # maximum difference between minimum crustal thickness
omega = pyshtools.constant.omega_mars.value
d_lith = 150.e3
d_sigma = 45.e3
t0 = 1.e3 # minimum crustal thickness
model = 10 # identifier for the interior reference model
# --- read 1D reference interior model ---
with open(interior_file[model], 'r') as f:
lines = f.readlines()
print(lines[0].strip())
data = lines[1].split()
if float(data[2]) != 1:
raise RuntimeError('Program not capable of reading polynomial ' +
'files')
num_file = int(lines[2].split()[0])
crust_index_file = int(lines[2].split()[3])
core_index_file = int(lines[2].split()[2])
i_crust_file = crust_index_file - 1
i_core_file = core_index_file - 1
radius = np.zeros(num_file)
rho = np.zeros(num_file)
num = 0
for i in range(0, num_file-1):
data = lines[i+3].split()
rb = float(data[0])
rhob = float(data[1])
data = lines[i+4].split()
rt = float(data[0])
rhot = float(data[1])
if rb == rt:
if i == i_core_file:
i_core = num
if i == i_crust_file:
i_crust = num
else:
radius[num] = rb
rho[num] = (rhot + rhob) / 2.
num += 1
radius[num] = rt
rho[num] = 0. # the density above the surface is zero
num += 1
n = num - 1
radius = radius[:n+1]
rho = rho[:n+1]
r0_model = radius[n]
print('Surface radius of model (km) = {:f}'.format(r0_model / 1.e3))
for i in range(0, n+1):
if radius[i] <= (r0_model - d_lith) and \
radius[i+1] > (r0_model - d_lith):
if radius[i] == (r0_model - d_lith):
i_lith = i
elif (r0_model - d_lith) - radius[i] <= radius[i+1] -\
(r0_model - d_lith):
i_lith = i
else:
i_lith = i + 1
break
rho_mantle = rho[i_crust-1]
rho_core = rho[i_core-1]
print('Mantle density (kg/m3) = {:f}'.format(rho_mantle))
print('Mantle radius (km) = {:f}'.format(radius[i_crust]/1.e3))
print('Core density (kg/m3) = {:f}'.format(rho_core))
print('Core radius (km) = {:f}'.format(radius[i_core]/1.e3))
print('Assumed depth of lithosphere (km) = {:f}'.format(d_lith / 1.e3))
print('Actual depth of lithosphere in discretized model (km) = {:f}'
.format((r0_model - radius[i_lith]) / 1.e3))
# --- Compute gravity contribution from hydrostatic density interfaces ---
thickave = 44.e3 # initial guess of average crustal thickness
r_sigma = topo.r0 - thickave
rho_c = 2900.
if True:
# compute values for a planet that is completely fluid
hlm_fluid, clm_fluid, mass_model = \
HydrostaticShape(radius, rho, omega, potential.gm, potential.r0)
print('--- Hydrostatic potential coefficients for a fluid planet ---')
print('c20 = {:e}\nc40 = {:e}'.format(clm_fluid.coeffs[0, 2, 0],
clm_fluid.coeffs[0, 4, 0]))
print('--- Hydrostatic relief of surface for a fluid planet ---')
print('h20 = {:e}\nh40 = {:e}'.format(hlm_fluid[n].coeffs[0, 2, 0],
hlm_fluid[n].coeffs[0, 4, 0]))
hlm, clm_hydro, mass_model = \
HydrostaticShapeLith(radius, rho, i_lith, potential, topo, rho_c,
r_sigma, omega, lmax_hydro)
print('Total mass of model (kg) = {:e}'.format(mass_model))
print('% of J2 arising from beneath lithosphere = {:f}'
.format(clm_hydro.coeffs[0, 2, 0]/potential.coeffs[0, 2, 0] * 100.))
potential.coeffs[:, :lmax_hydro+1, :lmax_hydro+1] -= \
clm_hydro.coeffs[:, :lmax_hydro+1, :lmax_hydro+1]
# --- Constant density model ---
rho_c = 2900.
print('-- Constant density model --\nrho_c = {:f}'.format(rho_c))
tmin = 1.e9
thickave = 44.e3 # initial guess of average crustal thickness
while abs(tmin - t0) > t0_sigma:
# iterate to fit assumed minimum crustal thickness
moho = pyMoho.pyMoho(potential, topo, lmax, rho_c, rho_mantle,
thickave, filter_type=filter, half=half,
lmax_calc=lmax_calc, nmax=nmax, quiet=True)
thick_grid = (topo.pad(lmax) - moho.pad(lmax)).expand(grid='DH2')
print('Average crustal thickness (km) = {:f}'.format(thickave / 1.e3))
print('Crustal thickness at InSight landing sites (km) = {:f}'
.format((topo.pad(lmax) - moho.pad(lmax))
.expand(lat=lat_insight, lon=lon_insight) / 1.e3))
tmin = thick_grid.min()
tmax = thick_grid.max()
print('Minimum thickness (km) = {:e}'.format(tmin / 1.e3))
print('Maximum thickness (km) = {:e}'.format(tmax / 1.e3))
thickave += t0 - tmin
thick_grid.plot(show=False, fname='Thick-Mars-1.png')
moho.plot_spectrum(show=False, fname='Moho-spectrum-Mars-1.png')
# --- Model with variable density ---
rho_south = 2900.
rho_north = 2900.
porosity = 0.0
print('-- Variable density model ---\n' +
'rho_south = {:f}\n'.format(rho_south) +
'rho_north = {:f}'.format(rho_north))
density = density * (rho_north - rho_south)
density.coeffs[0, 0, 0] += rho_south
tmin = 1.e9
thickave = 44.e3 # initial guess of average crustal thickness
while abs(tmin - t0) > t0_sigma:
# iterate to fit assumed minimum crustal thickness
moho = pyMoho.pyMohoRho(potential, topo, density, porosity, lmax,
rho_mantle, thickave, filter_type=filter,
half=half, lmax_calc=lmax_calc, quiet=True,
nmax=nmax)
thick_grid = (topo.pad(lmax) - moho.pad(lmax)).expand(grid='DH2')
print('Average crustal thickness (km) = {:e}'.format(thickave / 1.e3))
print('Crustal thickness at InSight landing sites (km) = {:e}'
.format((topo.pad(lmax) - moho.pad(lmax))
.expand(lat=lat_insight, lon=lon_insight) / 1.e3))
tmin = thick_grid.data.min()
tmax = thick_grid.data.max()
print('Minimum thickness (km) = {:e}'.format(tmin / 1.e3))
print('Maximum thickness (km) = {:e}'.format(tmax / 1.e3))
thickave += t0 - tmin
thick_grid.plot(show=False, fname='Thick-Mars-2.png')
moho.plot_spectrum(show=False, fname='Moho-spectrum-Mars-2.png')
def main():
gravfile = 'Data/gmm3_120_sha.tab'
topofile = 'Data/MarsTopo719.shape'
densityfile = 'Data/dichotomy_359.sh'
model_name = [
'dwcold', 'dwhot_modif', 'dwhot_stairs', 'DWTh2Ref1', 'DWTh2Ref2',
'eh70cold', '"eh70hot', 'Gudkova'
]
spec = 'Data/Mars-reference-interior-models/model_'
interior_file = [spec + name for name in model_name]
lmax_calc = 90
lmax = lmax_calc * 4
potcoefs, lmaxp, header = pyshtools.shio.shread(gravfile,
header=True,
lmax=lmax)
potential = pyshtools.SHCoeffs.from_array(potcoefs)
potential.r_ref = float(header[0]) * 1.e3
potential.gm = float(header[1]) * 1.e9
potential.mass = potential.gm / float(pyshtools.constant.grav_constant)
print('Gravity file = {:s}'.format(gravfile))
print('Lmax of potential coefficients = {:d}'.format(lmaxp))
print('Reference radius (km) = {:f}'.format(potential.r_ref / 1.e3))
print('GM = {:e}'.format(potential.gm))
print('Mass = {:e}'.format(potential.mass))
model = 3
lmax_hydro = 15
omega = float(pyshtools.constant.omega_mars)
print('Omega = {:e}'.format(omega))
# --- read 1D reference interior model ---
print('=== Reading model {:s} ==='.format(model_name[model]))
with open(interior_file[model], 'r') as f:
lines = f.readlines()
print(lines[0].strip())
data = lines[1].split()
if float(data[2]) != 1:
raise RuntimeError('Program not capable of reading polynomial ' +
'files')
num = int(lines[2].split()[0])
crust_index = int(lines[2].split()[3])
mantle_index = int(lines[2].split()[3])
radius = np.zeros(num)
rho = np.zeros(num)
for i in range(0, num):
data = lines[i + 3].split()
radius[i] = float(data[0])
rho[i] = float(data[1])
r0_model = radius[num - 1]
print('Surface radius of model (km) = {:f}'.format(r0_model / 1.e3))
n = num - 1
rho[n] = 0. # the density above the surface is zero
rho_mantle = rho[crust_index - 1]
print('Mantle density (kg/m3) = {:f}'.format(rho_mantle))
# --- Compute gravity contribution from hydrostatic density interfaces ---
if True:
for i in range(1, 8):
# compute values for a planet that is completely fluid
fn = True
nmax = i
hlm_fluid, clm_fluid, mass_model = \
HydrostaticShape(radius, rho, omega, potential.gm,
potential.r_ref, nmax=nmax,
finiteamplitude=fn)
print('--- Hydrostatic relief of surface ---')
print('nmax = {:d}'.format(nmax))
print('h20 = {:e}\n'.format(hlm_fluid[n].coeffs[0, 2, 0]) +
'h40 = {:e}'.format(hlm_fluid[n].coeffs[0, 4, 0]))
def main():
lmax = 20
lmax_grid = 719
omega = pyshtools.constant.omega_moon.value
rem = pyshtools.constant.a_orbit_moon.value
mass_earth = pyshtools.constant.mass_egm2008.value
cthick = 34.e3 # 43.e3 or 34.0e3
rho_crust = 2550.
out_rc_fc = "figs/rc_fc_34_2550.dat"
out_rc_rhoc = "figs/rc_rhoc_34_2550.dat"
out_rc_beta = "figs/rc_beta_34_2550.dat"
sh_core = "figs/core_34.sh"
core_shape_wo_d1 = "figs/core_shape_wo_d1_330_34.dat"
core_shape = "figs/core_shape_330_34.dat"
rcore_int = 1.e3
rcore_start = 250.e3
rcore_end = 450.e3
rhocore_start = 5000.
rhocore_end = 8000.
rhocore_int = 1.
pot_file = 'Data/JGGRAIL_900C11A_SHA.TAB'
topo_file = 'Data/LOLA1500p.sh'
potential = pyshtools.SHGravCoeffs.from_file(pot_file, header_units='km')
topo = pyshtools.SHCoeffs.from_file(topo_file, lmax=900)
r0 = topo.coeffs[0, 0, 0]
r_sigma = r0 - cthick
ismr2 = 0.3927280 # Williams et al. (2014)
ismr2 = ismr2 * (1738.e3 / r0)**2
print("Mean planetary radius (km) = {:e}".format(r0 / 1.e3))
print("Is/MR2 (solid Moon using mean radius) = {:e}".format(ismr2))
print("Lmax of Gravitational potential = {:d}".format(potential.lmax))
print("Reference radius of potential model (km) = {:e}".format(
potential.r0 / 1.e3))
print("GM = {:e}".format(potential.gm))
mass = potential.gm / pyshtools.constant.G.value
print("Mass (kg) = {:e}".format(mass))
print("Omega = {:e}".format(omega))
print("Period (days) = {:e}".format(2. * np.pi / omega / 60. / 60. / 24.))
print("Average crustal thickness (km) = {:e}".format(cthick / 1.e3))
print("Crustal density (kg/m3) = {:e}".format(rho_crust))
radius = np.zeros(4)
radius[0] = 0.
radius[2] = r0 - cthick
radius[3] = r0
rho = np.zeros(4)
rho[2] = rho_crust
mass_crust = 4. * np.pi / 3. * rho_crust * (radius[3]**3 - radius[2]**3)
n = 3
i_lith = 2
i_core = 1
# For each core radius, find rho_mantle and rho_core that fit total mass
# and moment of inertia
f_rc_fc = open(out_rc_fc, 'w')
f_rc_rhoc = open(out_rc_rhoc, 'w')
f_rc_beta = open(out_rc_beta, 'w')
for r_core in np.arange(rcore_start,
rcore_end + rcore_int,
rcore_int,
dtype=float):
radius[1] = r_core
first = True
for rho_core in np.arange(rhocore_start,
rhocore_end + rhocore_int,
rhocore_int,
dtype=float):
mass_core = 4. * np.pi / 3. * rho_core * r_core**3
mass_mantle = mass - mass_crust - mass_core
rho_mantle = mass_mantle * 3. / np.pi / 4. / (radius[2]**3 -
r_core**3)
rho[0] = rho_core
rho[1] = rho_mantle
if rho_mantle >= rho_core:
continue
ismr2_model = moi_solid(radius, rho, n)
if first is True:
diff_old = ismr2 - ismr2_model
first = False
else:
diff_new = ismr2 - ismr2_model
if diff_new * diff_old <= 0.:
# interpolate to get the best fitting core density
rho_core_final = (rho_core - rhocore_int) - diff_old * \
(rhocore_int) / (diff_new - diff_old)
rho[0] = rho_core_final
mass_core = 4. * np.pi / 3. * rho[0] * r_core**3
mass_mantle = mass - mass_crust - mass_core
rho_mantle = mass_mantle * 3. / np.pi / 4. / \
(radius[2]**3 - r_core**3)
rho[1] = rho_mantle
hlm, clm_hydro, mass_model = \
HydrostaticShapeLith(radius, rho, i_lith,
potential, topo, rho_crust,
r_sigma, omega, lmax,
rp=rem, mp=mass_earth)
# set degree-1 terms to zero
hlm[1].coeffs[:, 1, :] = 0.
a = hlm[1].expand(lat=0., lon=0., lmax_calc=lmax)
b = hlm[1].expand(lat=0., lon=90., lmax_calc=lmax)
c = hlm[1].expand(lat=90., lon=0., lmax_calc=lmax)
f_core = ((a + b) / 2. - c) / ((a + b) / 2.)
beta_core = (a**2 - b**2) / (a**2 + b**2)
print(r_core / 1.e3, rho[0], rho[1], f_core, beta_core)
f_rc_fc.write('{:e}, {:e}\n'.format(r_core / 1.e3, f_core))
f_rc_rhoc.write('{:e}, {:e}\n'.format(
r_core / 1.e3, rho[0]))
f_rc_beta.write('{:e}, {:e}\n'.format(
r_core / 1.e3, beta_core))
if r_core == 330.e3:
print("Rcore (km) = {:e}".format(r_core / 1.e3))
print("A (km) = {:e}".format(a / 1.e3))
print("B (km) = {:e}".format(b / 1.e3))
print("C (km) = {:e}".format(c / 1.e3))
print("rho_core (kg/m3) = {:e}".format(rho[0]))
II, AA, BB, CC, mass_model, RR, vec = \
InertiaTensor_from_shape(hlm, rho, 1, quiet=False)
grid = hlm[i_core].expand(lmax=lmax_grid, grid='DH2')
grid.to_file(core_shape_wo_d1)
print("Size of output grid = {:d}, {:d}".format(
grid.nlat, grid.nlon))
print("Maximum = {:e}\nMinimum = {:e}".format(
grid.max(), grid.min()))
hlm, clm_hydro, mass_model = \
HydrostaticShapeLith(radius, rho, i_lith,
potential, topo, rho_crust,
r_sigma, omega, lmax,
rp=rem, mp=mass_earth)
hlm_fluid, clm_fluid, mass_model = \
HydrostaticShape(radius, rho, omega, potential.gm,
potential.r0,
rp=rem, mp=mass_earth)
grid = hlm[i_core].expand(lmax=lmax_grid, grid='DH2')
grid_fluid = hlm_fluid[i_core].expand(lmax=lmax_grid,
grid='DH2')
a_fluid = hlm_fluid[1].expand(lat=0., lon=0.)
b_fluid = hlm_fluid[1].expand(lat=0., lon=90.)
c_fluid = hlm_fluid[1].expand(lat=90., lon=0.)
f_core_fluid = (((a_fluid + b_fluid) / 2. - c_fluid) /
((a_fluid + b_fluid) / 2.))
beta_core_fluid = ((a_fluid**2 - b_fluid**2) /
(a_fluid**2 + b_fluid**2))
print('f_core for a fluid planet = ', f_core_fluid)
print('beta_core for a fluid planet = ',
beta_core_fluid)
diff = grid - grid_fluid
print(
'Maximuim and miniumum core relief for '
'a fluid planet (m) = ', grid_fluid.max(),
grid_fluid.min())
print(
'Maximuim and miniumum difference with respect '
'to a fluid planet (m) = ', diff.max(), diff.min())
hlm[i_core].to_file(sh_core)
grid.to_file(core_shape)
print("Maximum = {:e}\nMinimum = {:e}".format(
grid.max(), grid.min()))
for l in range(0, 4):
for m in range(0, l + 1):
print(l, m, hlm[i_core].coeffs[0, l, m],
hlm[i_core].coeffs[1, l, m])
diff_old = diff_new
def main():
gravfile = 'Data/gmm3_120_sha.tab'
topofile = 'Data/MarsTopo719.shape'
densityfile = 'Data/dichotomy_359.sh'
model_name = [
'DWThot', 'DWThotCrust1', 'DWThotCrust1r', 'EH45Tcold',
'EH45TcoldCrust1', 'EH45TcoldCrust1r', 'EH45ThotCrust2',
'EH45ThotCrust2r', 'LFAK', 'SANAK', 'TAYAK', 'DWAK', 'ZG_DW'
]
spec = 'Data/Mars-reference-interior-models/Smrekar/'
interior_file = [spec + name + '.deck' for name in model_name]
lmax_calc = 90
lmax = lmax_calc * 4
potential = pyshtools.SHGravCoeffs.from_file(gravfile, header_units='km')
print('Gravity file = {:s}'.format(gravfile))
print('Lmax of potential coefficients = {:d}'.format(potential.lmax))
print('Reference radius (km) = {:f}'.format(potential.r0 / 1.e3))
print('GM = {:e}\n'.format(potential.gm))
topo = pyshtools.SHCoeffs.from_file(topofile, lmax=lmax)
topo.r0 = topo.coeffs[0, 0, 0]
print('Topography file = {:s}'.format(topofile))
print('Lmax of topography coefficients = {:d}'.format(topo.lmax))
print('Reference radius (km) = {:f}\n'.format(topo.r0 / 1.e3))
density = pyshtools.SHCoeffs.from_file(densityfile, lmax=lmax)
print('Lmax of density coefficients = {:d}\n'.format(density.lmax))
lat_insight = 4.502384
lon_insight = 135.623447
filter = 1
half = 50
nmax = 7
lmax_hydro = 15
t0_sigma = 5. # maximum difference between minimum crustal thickness
omega = pyshtools.constant.omega_mars.value
d_lith = 150.e3
d_sigma = 45.e3
t0 = 1.e3 # minimum crustal thickness
model = 10 # identifier for the interior reference model
# --- read 1D reference interior model ---
radius, rho, i_crust, i_core, i_lith = ReadRefModel(interior_file[model],
depth=d_lith,
quiet=False)
rho_mantle = rho[i_crust - 1]
rho_core = rho[i_core - 1]
n = len(radius) - 1
r0_model = radius[n]
# --- Compute gravity contribution from hydrostatic density interfaces ---
thickave = 44.e3 # initial guess of average crustal thickness
r_sigma = topo.r0 - thickave
rho_c = 2900.
if True:
# compute values for a planet that is completely fluid
hlm_fluid, clm_fluid, mass_model = \
HydrostaticShape(radius, rho, omega, potential.gm, potential.r0)
print('--- Hydrostatic potential coefficients for a fluid planet ---')
print('c20 = {:e}\nc40 = {:e}'.format(clm_fluid.coeffs[0, 2, 0],
clm_fluid.coeffs[0, 4, 0]))
print('--- Hydrostatic relief of surface for a fluid planet ---')
print('h20 = {:e}\nh40 = {:e}'.format(hlm_fluid[n].coeffs[0, 2, 0],
hlm_fluid[n].coeffs[0, 4, 0]))
hlm, clm_hydro, mass_model = \
HydrostaticShapeLith(radius, rho, i_lith, potential, topo, rho_c,
r_sigma, omega, lmax_hydro)
print('Total mass of model (kg) = {:e}'.format(mass_model))
print('% of J2 arising from beneath lithosphere = {:f}'.format(
clm_hydro.coeffs[0, 2, 0] / potential.coeffs[0, 2, 0] * 100.))
potential.coeffs[:, :lmax_hydro+1, :lmax_hydro+1] -= \
clm_hydro.coeffs[:, :lmax_hydro+1, :lmax_hydro+1]
# --- Constant density model ---
rho_c = 2900.
print('-- Constant density model --\nrho_c = {:f}'.format(rho_c))
tmin = 1.e9
thickave = 44.e3 # initial guess of average crustal thickness
while abs(tmin - t0) > t0_sigma:
# iterate to fit assumed minimum crustal thickness
moho = pyMoho.pyMoho(potential,
topo,
lmax,
rho_c,
rho_mantle,
thickave,
filter_type=filter,
half=half,
lmax_calc=lmax_calc,
nmax=nmax,
quiet=True)
thick_grid = (topo.pad(lmax) - moho.pad(lmax)).expand(grid='DH2')
print('Average crustal thickness (km) = {:f}'.format(thickave / 1.e3))
print('Crustal thickness at InSight landing sites (km) = {:f}'.format(
(topo.pad(lmax) - moho.pad(lmax)).expand(lat=lat_insight,
lon=lon_insight) / 1.e3))
tmin = thick_grid.min()
tmax = thick_grid.max()
print('Minimum thickness (km) = {:e}'.format(tmin / 1.e3))
print('Maximum thickness (km) = {:e}'.format(tmax / 1.e3))
thickave += t0 - tmin
thick_grid.plot(show=False, fname='Thick-Mars-1.png')
moho.plot_spectrum(show=False, fname='Moho-spectrum-Mars-1.png')
# --- Model with variable density ---
rho_south = 2900.
rho_north = 2900.
porosity = 0.0
print('-- Variable density model ---\n' +
'rho_south = {:f}\n'.format(rho_south) +
'rho_north = {:f}'.format(rho_north))
density = density * (rho_north - rho_south)
density.coeffs[0, 0, 0] += rho_south
tmin = 1.e9
thickave = 44.e3 # initial guess of average crustal thickness
while abs(tmin - t0) > t0_sigma:
# iterate to fit assumed minimum crustal thickness
moho = pyMoho.pyMohoRho(potential,
topo,
density,
porosity,
lmax,
rho_mantle,
thickave,
filter_type=filter,
half=half,
lmax_calc=lmax_calc,
quiet=True,
nmax=nmax)
thick_grid = (topo.pad(lmax) - moho.pad(lmax)).expand(grid='DH2')
print('Average crustal thickness (km) = {:e}'.format(thickave / 1.e3))
print('Crustal thickness at InSight landing sites (km) = {:e}'.format(
(topo.pad(lmax) - moho.pad(lmax)).expand(lat=lat_insight,
lon=lon_insight) / 1.e3))
tmin = thick_grid.data.min()
tmax = thick_grid.data.max()
print('Minimum thickness (km) = {:e}'.format(tmin / 1.e3))
print('Maximum thickness (km) = {:e}'.format(tmax / 1.e3))
thickave += t0 - tmin
thick_grid.plot(show=False, fname='Thick-Mars-2.png')
moho.plot_spectrum(show=False, fname='Moho-spectrum-Mars-2.png')