def transfer(fs, ts, khat, xyz, L=5e9):
khat = tt.reshape(khat, (1, 3))
fstar = 2.99792e8 / (2 * pi * L)
tfs_re = tt.zeros((3, 3, ts.shape[0]))
tfs_im = tt.zeros((3, 3, ts.shape[0]))
for i in range(3):
for j in range(i + 1, 3):
if not (j == i):
rij = xyz[j, :, :] - xyz[i, :, :]
r2 = tt.reshape(pmm.sum(rij * rij, axis=1), (-1, 1))
rij = rij / pmm.sqrt(r2)
rk = pmm.sum(rij * khat, axis=1)
w = fs / (2 * fstar) * (1 - rk)
wp = fs / (2 * fstar) * (1 + rk
) # Reverse direction, get a plus
sc = pmm.sin(w) / w
scp = pmm.sin(wp) / wp
tfs_re = tt.set_subtensor(tfs_re[i, j, :], sc * pmm.cos(w))
tfs_im = tt.set_subtensor(tfs_im[i, j, :], sc * pmm.sin(w))
tfs_re = tt.set_subtensor(tfs_re[j, i, :], scp * pmm.cos(wp))
tfs_im = tt.set_subtensor(tfs_im[j, i, :], scp * pmm.sin(wp))
return tfs_re, tfs_im
def constellation(ts, e=0.00985, R=1, kappa=0, lam=0):
L = 5e9 * 2 * sqrt(3) * e * R # m ; R in AU
alpha = 2 * pi * ts / (3600.0 * 24.0 * 365.25 * R**(3.0 / 2.0)) + kappa
alpha = tt.reshape(alpha, (1, alpha.shape[0]))
betas = tt.as_tensor_variable(
[kappa, kappa + 2 * pi / 3, kappa + 4 * pi / 3])
betas = tt.reshape(betas, (betas.shape[0], 1))
xs = R * pmm.cos(alpha) + 0.5 * e * R * (pmm.cos(2 * alpha - betas) -
3 * pmm.cos(betas))
ys = R * pmm.sin(alpha) + 0.5 * e * R * (pmm.sin(2 * alpha - betas) -
3 * pmm.sin(betas))
zs = -sqrt(3) * e * R * pmm.cos(alpha - betas)
return tt.stack((xs, ys, zs), axis=2)
def y_slow(ts, f0, fdot, fddot, phi0, nhat, cos_iota, psi, fhet=None):
if fhet is None:
fhet = f0
df = f0 - fhet
uvk_ = uvk(nhat)
khat = uvk_[2, :]
xyz = constellation(ts)
dp, dc = dp_dc(xyz, uvk_)
cp = pmm.cos(2.0 * psi)
sp = pmm.sin(2.0 * psi)
ys_re = tt.zeros((3, 3, ts.shape[0]))
ys_im = tt.zeros((3, 3, ts.shape[0]))
for i in range(3):
kdotx = 499.0048 * tt.tensordot(
xyz[i, :, :], khat, axes=1) # 499.0048 = 1 AU / c in seconds
xi = ts - kdotx
fs = f0 + fdot * xi
tfs_re, tfs_im = transfer(fs, ts, khat, xyz)
Ap, Ac = Ap_Ac(fs, cos_iota)
phi = pi * fddot * xi * xi * xi / 3.0 + pi * fdot * xi * xi + phi0 - 2 * pi * f0 * kdotx + 2 * pi * df * ts
osc_re = pmm.cos(phi)
osc_im = pmm.sin(phi)
for j in range(3):
if not i == j:
Fp_re = dp[i, j] * Ap * cp - dc[i, j] * Ap * sp
Fp_im = -dp[i, j] * Ac * sp - dc[i, j] * Ac * cp
A_re = (Fp_re * tfs_re[i, j, :] - Fp_im * tfs_im[i, j, :]) / 4
A_im = (Fp_im * tfs_re[i, j, :] + Fp_re * tfs_im[i, j, :]) / 4
ys_re = tt.set_subtensor(ys_re[i, j, :],
A_re * osc_re - A_im * osc_im)
ys_im = tt.set_subtensor(ys_im[i, j, :],
A_im * osc_re + A_re * osc_im)
return (ys_re, ys_im)
def uvk(nhat):
phi = tt.arctan2(nhat[1], nhat[0])
cos_theta = nhat[2]
sin_theta = pmm.sqrt(1 - cos_theta * cos_theta)
cphi = pmm.cos(phi)
sphi = pmm.sin(phi)
u = tt.as_tensor_variable([cos_theta * cphi, cos_theta * sphi, -sin_theta])
v = tt.as_tensor_variable([sphi, -cphi, 0.0])
k = tt.as_tensor_variable(
[-sin_theta * cphi, -sin_theta * sphi, -cos_theta])
return tt.stack((u, v, k), dim=0)
def XYZ_freq(yf_re, yf_im, Tobs, heterodyne_bin, N, L=5e9):
fstar = 2.99792e8 / (2.0 * pi * L)
nf = N
df = 1.0 / Tobs
dfs = linspace(0, df * nf / 2, int(round(nf / 2)) + 1)
dfs = concatenate((dfs, -dfs[-2:0:-1]))
fs = heterodyne_bin * df + dfs
c1 = pmm.cos(-fs / fstar)
s1 = pmm.sin(-fs / fstar)
c2 = pmm.cos(-2 * fs / fstar)
s2 = pmm.sin(-2 * fs / fstar)
c3 = pmm.cos(-3 * fs / fstar)
s3 = pmm.sin(-3 * fs / fstar)
X_re = yf_re[0,1,:]*c3 - yf_im[0,1,:]*s3 \
- (yf_re[0,2,:]*c3 - yf_im[0,2,:]*s3) \
+ yf_re[1,0,:]*c2 - yf_im[1,0,:]*s2 \
- (yf_re[2,0,:]*c2 - yf_im[2,0,:]*s2) \
+ yf_re[0,2,:]*c1 - yf_im[0,2,:]*s1 \
- (yf_re[0,1,:]*c1 - yf_im[0,1,:]*s1) \
+ yf_re[2,0,:] - yf_re[1,0,:]
X_im = yf_im[0,1,:]*c3 + yf_re[0,1,:]*s3 \
- (yf_im[0,2,:]*c3 + yf_re[0,2,:]*s3) \
+ yf_im[1,0,:]*c2 + yf_re[1,0,:]*s2 \
- (yf_im[2,0,:]*c2 + yf_re[2,0,:]*s2) \
+ yf_im[0,2,:]*c1 + yf_re[0,2,:]*s1 \
- (yf_im[0,1,:]*c1 + yf_re[0,1,:]*s1) \
+ yf_im[2,0,:] - yf_im[1,0,:]
Y_re = yf_re[1,2,:]*c3 - yf_im[1,2,:]*s3 \
- (yf_re[1,0,:]*c3 - yf_im[1,0,:]*s3) \
+ yf_re[2,1,:]*c2 - yf_im[2,1,:]*s2 \
- (yf_re[0,1,:]*c2 - yf_im[0,1,:]*s2) \
+ yf_re[1,0,:]*c1 - yf_im[1,0,:]*s1 \
- (yf_re[1,2,:]*c1 - yf_im[1,2,:]*s1) \
+ yf_re[0,1,:] - yf_re[2,1,:]
Y_im = yf_im[1,2,:]*c3 + yf_re[1,2,:]*s3 \
- (yf_im[1,0,:]*c3 + yf_re[1,0,:]*s3) \
+ yf_im[2,1,:]*c2 + yf_re[2,1,:]*s2 \
- (yf_im[0,1,:]*c2 + yf_re[0,1,:]*s2) \
+ yf_im[1,0,:]*c1 + yf_re[1,0,:]*s1 \
- (yf_im[1,2,:]*c1 + yf_re[1,2,:]*s1) \
+ yf_im[0,1,:] - yf_im[2,1,:]
Z_re = yf_re[2,0,:]*c3 - yf_im[2,0,:]*s3 \
- (yf_re[2,1,:]*c3 - yf_im[2,1,:]*s3) \
+ yf_re[0,2,:]*c2 - yf_im[0,2,:]*s2 \
- (yf_re[1,2,:]*c2 - yf_im[1,2,:]*s2) \
+ yf_re[2,1,:]*c1 - yf_im[2,1,:]*s1 \
- (yf_re[2,0,:]*c1 - yf_im[2,0,:]*s1) \
+ yf_re[1,2,:] - yf_re[0,2,:]
Z_im = yf_im[1,2,:]*c3 + yf_re[1,2,:]*s3 \
- (yf_im[1,0,:]*c3 + yf_re[1,0,:]*s3) \
+ yf_im[2,1,:]*c2 + yf_re[2,1,:]*s2 \
- (yf_im[0,1,:]*c2 + yf_re[0,1,:]*s2) \
+ yf_im[1,0,:]*c1 + yf_re[1,0,:]*s1 \
- (yf_im[1,2,:]*c1 + yf_re[1,2,:]*s1) \
+ yf_im[0,1,:] - yf_im[2,1,:]
return ((X_re, X_im), (Y_re, Y_im), (Z_re, Z_im))
def distance_f():
return D.d('v_mean') * cos(D.d('theta')) * \
(D.d('v_mean') * sin(D.d('theta')) + sqrt((D.d('v_mean') * sin(D.d('theta'))) ** 2 +\
2 * g * D.d('height')) / g)