 | One dimensional gaussian probability function
is
g(x) = 1/sigma/sqrt(2PI) *
exp[-(x-x0)^2/2/sigma^2],
where sigma is the standard diviation of the
gaussian probability distribution. The FWHM
(=xb) is
FWHM = xb = 2sqrt(2ln2) *
sigma.
The gaussian function can be expressed in terms of FWHM as
g(x) = 2*sqrt(ln2/PI)/xb * exp[-4ln2 * (x-x0)^2/xb^2]. |
| Two dimensional circular gaussian function
(standard deviation sigma_x = sigma_y = sigma) is
g(x,y) = 1/sigma^2/2/PI *
exp{-[(x-x0)^2+(y-y0)^2]/2/sigma^2},
or expressed in radius r^2 = (x-x0)^2+(y-y0)^2 as
g(r) = 1/sigma^2/2/PI * exp[-r^2/2/sigma^2].
Elliptical gaussian function (sigma_x !=
sigma_y) is
g(x,y) = 1/sigma_x/sigma_y/2/PI *
exp{-[(x-x0)^2/2/sigma_x^2+(y-y0)^2/2/sigma_y^2]}. |
| Coupling of
a gaussian beam with a gaussian source. When an antenna with
normallized gaussian beam (FWHM=theta_b)
Pn(r,theta) = exp(-4ln2 * theta^2/theta_b^2)
is pointed to the center of a gaussian source (FWHM = theta_s)
B(theta,T) = J(T) * exp(-4ln2 * theta^2/theta_s^2),
the observed antenna main beam temperature T_mb is equal to brightness
temperature at the source center multiplied by a beam dilution factor f.
The value of f can be derived by
Flux = int{B(theta,T)*Pn(r,theta)} * 2PI * theta * d
theta | theta = 0-> inf
= J(T)
int{exp[-4ln2 * theta^2 * (1/theta_s^2 + 1/theta_b^2)]} * 2PI * theta * d
theta | theta = 0-> inf
=
PI/4/ln2 * J(T)*theta_s^2/(theta_s^2+theta_b^2) * theta_b^2
def=
PI/4/ln2 * T_mb * theta_b^2
The last equation is equivalent to a definition of the main beam antenna
temperature T_mb. Therefore, the beam dilution factor
f = T_mb / J(T) = thetas^2 / (theta_s^2+theta_b^2). |
| Coupling of
a gaussian beam and a disk source. When an antenna with normallized
gaussian beam (FWHM=theta_b)
Pn(r,theta) = exp(-4ln2 * theta^2/theta_b^2)
is pointed to the center of a homogeneous disk source with brightness
temperature J(T) (diameter = theta_s), the observed antenna main beam
temperature T_mb is equal to brightness temperature multiplied by a beam
dilution factor f. The value of f can be derived by
Flux = int{J(T)*Pn(r,theta)} * 2PI * theta * d theta |
theta = 0->theta_s/2
= J(T)
int{exp[-4ln2 * theta^2/theta_b^2)]} * 2PI * theta * d theta | theta =
0-> theta_s/2
=
PI/4/ln2 * J(T)*[1-exp(-ln2 * theta_s^2/theta_b^2)] * theta_b^2
def=
PI/4/ln2 * T_mb * theta_b^2
The last equation is equivalent to a definition of the main beam antenna
temperature T_mb. Therefore, the beam dilution factor
f = T_mb / J(T) = 1-exp(-ln2 * theta_s^2/theta_b^2). |
Convert thermal gas temperature into average
particle thermal velocity
The average kinetic energy of a particle in a thermal gas is
E_k = 3/2 * kT = 1/2 * m *
V_th^2
where k is Boltzmann constant, T is the thermal temperature, m is the mass of
the particle and V_th is the thermal velocity of the particle. Therefore,
V_th =
sqrt [ 3kT / m ] ~ sqrt [ (T/100K)
/ (n_p) ] * 1.5745 km/s.
Here n_p is the mass number of the particle -- the mass devided by the particle
mass of hydrogen mass: n_p = m / m_H.
Example 1: In room temperature T_k = 300 K, V_th (H2) ~ 1.9
km/s, V_th (N2) ~ 500 m/s, V_th (O2) ~ 482 m/s.
Example 2: In interstellar medium, with the typical T_k = 10000 K, V_th (H2)
~ 16 km/s.
Example 3: In dark cloud, with the typical T_k = 10 K, V_th (H2) ~ 0.35 km/s.
Example 4: In molecular cloud, with the typical T_k = 10 K, V_th (H2) ~
0.35 km/s, V_th (CO) ~ 0.1
km/s.
Example 5: In circumstellar envelope, with the typical T_k = 100 K, V_th (H2)
~ 1.1 km/s, V_th (CO) ~ 0.3
km/s.
ynao.ac.cn
[back
to my Home]