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Файл:Антарктида (3 сем, мага) / HW1 / AinA_HW01
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Astrophysics in Antarctica Homework 01 October 7, 2023
The Fiis Transmission Equation:
P
R
P
T
= GTG
R
2
λ
. (1)
4πl
The Friis Transmission Equation relates the power received PRto the power transmitted PTbetween two antennas separated by a distance L. The term (λ/4πl)2is called the free-space loss
factor, and it takes into account the losses due to the spherical spreading of the energy by the
antenna. GTand GRare the antenna gains (with respect to an isotropic radiator) of the transmitting and receiving antennas respectively, λ is the wavelength representing the effective aperture
area of the receiving antenna.
In our case, we should also take into consideration the factor e
−l/L
att
(I = I0e
−µR
= I0e
−l/L
att
which relates to the power loss due to absorption of waves in the ice, and the power reflection
coefficient R
ref lection
(reflection at the ice-water interface):
2
PR= GTGRP
λ
T
4πl
e
−l/L
att
R
ref lection
. (2)
The distance between the antennas l (the total transit distance) is twice the ice depth (l = 2z, z =
50 km), L
is the attenuation length over the frequency range of interest, GTand GRare the
att
gains of transmitter and receiver in-ice (GT= GR= 10). The reflection coefficient:
√
R
ref lection
=
√
ε
ε
water
water
−√ε
+√ε
ice
, (3)
ice
),
where ε
water
and ε
- the complex dielectric constants of the water and ice correspondingly.
ice
The signal can be observed only if it’s above the threshold. The threshold is associated with
the thermal noise:
P
= kBTSB, (4)
therm
where B is a bandwidth (f = 300 ± 1 MHz, B = 1 MHz), TS= 80 K is the temperature on the
surface of Enceladus and kB= 1.38 ·10
−23m2
· kg · s−2· K−1.
The signal-to-noise ratio: S : N ≥ 5. Hence, the power received is given by:
PR≥ (S : N ) ·P
= 5kBTSB. (5)
therm
The power transmitted is given by
PT=
P
R
GTG
R
4π ·2z
λ
2
2z/L
· e
att
5kBTSB
=
GTG
R
4π ·2z
2
2z/L
· e
λ
att
√
√
ε
water
ε
water
−√ε
+√ε
ice
. (6)
ice
Therefore, the amplitude of the transmitted signal VT:
2
V
PT=
T
=⇒ VT=pPTR,
R
VT=
5kBTSB
GTG
√
ε
+√ε
water
R
√
ε
R
water
−√ε
ice
ice
!
1/2
8πz
λ
z/L
att
· e
, (7)
1
Astrophysics in Antarctica Homework 01 October 7, 2023
where R = 50 Ω.
With frequency f = 300 MHz the wavelength is
λ = c/f =
300 ·106s
= 1 m (8)
−1
The time delay of the sigal:
3 ·108m/s
n
∆t =
c
n
l =
2z, (9)
c
where c is the speed of light, n is the refractive index of ice, z is the ice depth.
Complex dielectric constant:
ε∗= ε′+ iε′′= ε′(1 −i ·tg(δ)), (10)
where tg(δ) is the loss tagent and n =√ε′=pRe(ε) is the refractive index.
The refractive index can be also calculated more precisely using:
n2=
1
(√ε′2+ ε
′′2
+ ε′). (11)
2
The attenuation length:
Barwick, S., Besson, D., Gorham, P., Saltzberg, D. (2005). South Polar in situ radio-frequency ice
attenuation. Journal of Glaciology, 51(173), 231-238. doi:10.3189/172756505781829467
′
att
∼
1fε
, (12)
′′
ε
L
where f - the frequency. In the frequency range 0.1 − 2 GHz the attenuation length is expected
to remain constant: ε′′∼ 1/f, L
∼ ε′∼ const.
att
The attenuation length is inverse of the loss, which is related to the loss tagent tg(δ) = ε′′/ε
via:
loss[dB · m−1] = 8.686
√
ω
ε′tg(δ), (13)
2c
where ω = 2πf is the angular frequency. Hence, the attenuation length:
att
=
µ[m−1]
=
0.23 ·loss[dB ·m−1]
L
1
1
=
c
[m]. (14)
0.23 ·8.686πfpε′tg(δ)
The Temperature-Depth Profile
Ruiz, J., Fair´en, A.G. Seas under ice: Stability of liquid-water oceans within icy worlds. Earth Moon
Planet 97, 79–90 (2005). https://doi.org/10.1007/s11038-005-9052-8
′
2
Astrophysics in Antarctica Homework 01 October 7, 2023
If we treat the outer ice shell as a horizontal, ”flat”, layer and take into account that the thermal
conductivity of water ice is a function of temperature as k = k0/T (where k0= 567 W·m−1;
Klinger, 1980), then the temperature profile in the shell can be calculated using:
Tz= Tsexp
zF
, (15)
k
0
where Tsis the surface temperature, z is depth, and F is the vertical heat flow through the layer.
Figure 1 shows the temperature profile in an outer ice layer on Enceladus, calculated by means of
equation (18) assuming a surface temperature of 80 K (Ellsworth and Schubert, 1983) and a
surface heat flow of 5 mW·m−2(it is necessary remind that this heat flow value is an upper
limit). Ice melting temperature (assumed as 273 K), is reached at ∼ 140 km depth.
Figure 1: Temperature-depth profiles for Enceladus calculated for Cartesian and spherical geometries, assuming a vertical heat flow through the ice shell of ∼ 5 mW·m
−2
However, in a spherical shell in thermal conductive equilibrium heated from below, the vertical heat flow to a depth z is Fz= F r2/(r − z)2, where r is the body radius. In this case, the
temperature-depth profile of an icy body is given by
Tz= Tsexp
h
rF z
i
. (16)
k0(r − z)
If the temperature profile is calculated for a spherical geometry using equation (16) and a radius
for Enceladus of 250 km, the obtained curve is also shown in Figure 1. It can be clearly seen
that temperature increases more rapidly with depth when a spherical geometry is taken into account. In this case, the water ice melting temperature is reached at 90 km depth, which increases
the possibility of an internal liquid water layer for this small satellite.
In our task the ice depth z is 50 km. Let’s calculate the ice temperature on Enceladus
at this depth, assuming Ts= 80 K, F = 5 mW·m−2, r = 250 km, k0= 567 W·m−1.
3
Astrophysics in Antarctica Homework 01 October 7, 2023
Cartesian geometry:
50 ·103m ·5 · 10−3W ·m
Tz= 80 K · exp
567 W · m
Spherical geometry:
250 ·103m ·5 · 10−3W ·m−2· 50 ·103m
Tz= 80 K · exp
567 W · m−1(250 −50) · 103m
The dielectric constant depedence on temperature
−1
−2
= 124 K. (17)
= 139 K. (18)
′′
Figure 2: tg(δ) =
ε
dependence on temperature (V.Bogorodsky. Radioglaciology, 1985)
′
ε
This data is plotted and fitted by the exponential function (Figure 3).
The values of loss tangent at the mean temperature < T >= (Ts+ Tz)/2 are extracted
from plot:
• Iceland, the glacier ice:
– temperature range 80-124 K: < T >= 102 K, tg(δ) = 0.00000015;
– temperature range 80-139 K: < T >= 110 K, tg(δ) = 0.00000022.
• Little America, Antarctica:
– temperature range 80-124 K: < T >= 102 K, tg(δ) = 0.00000676;
– temperature range 80-139 K: < T >= 110 K, tg(δ) = 0.00000871.
4
Astrophysics in Antarctica Homework 01 October 7, 2023
′′
Figure 3: tg(δ) =
ε
dependence on temperature
′
ε
The loss tangent is also calculated by integrating the fit-function over the temperature
interval Ts− Tz:
• Iceland, the glacier ice:
– temperature range 80-124 K: tg(δ) = 0.00000018;
– temperature range 80-139 K: tg(δ) = 0.00000031.
• Little America, Antarctica:
– temperature range 80-124 K: tg(δ) = 0.00000732;
– temperature range 80-139 K: tg(δ) = 0.00000986.
The real part of the complex dielectric constant is estimated using data shown in Figure 4
(J.Appl. Phys., Vol.80, No.10, 15 November 1996 ).
Let’s assume that in our temperature and frequency range ε′≈ 3.12. Then the complex di-
electric constant of ice is ε
= 3.12(1 − 0.0000096i) and the refractive index of ice is
ice
n =√ε′=√3.12 ≈ 1.77.
The attenuation length (Eq. 14) was calculated using the value of loss tangent estimated
at the spherical geometry with data from:
• Little America, Antarctica
L
=
att
0.23 ·8.686 · π ·300 ·106Hz√3.12 ·0.00000986
3 ·108m/s
≈ 28.7 m; (19)
5
Astrophysics in Antarctica Homework 01 October 7, 2023
Figure 4: ε′dependence on temperature
• Iceland, the glacier ice
3 ·108m/s
L
=
att
0.23 ·8.686 · π ·300 ·106Hz√3.12 ·0.00000031
≈ 162.0 m. (20)
One more estimation of the attenuation length
The another estimation of the attenuation length was done using the plots, which was shown
on the lecture (Figure 5).
Figure 5: The ice depth-temperature dependence (left panel) and the attenuation length dependence on the depth of the ice (right panel)
6
Astrophysics in Antarctica Homework 01 October 7, 2023
These plots were digitized and the following data was extracted:
The L
ure 7).
Temperature,◦C Temperature, K Depth, m L
att
, m
-51 222 30 2017
-48 225 820 1665
-47 226 1000 1555
-44 229 1230 1404
-41 232 1500 1177
-36 237 1800 908
-31 242 2000 730
-26 247 2200 555
-18 255 2500 342
Figure 6: The temperature, depth and L
dependence on the temperature was plotted and fitted by the exponential function (Fig-
att
values extracted from plots on Figure 5
att
Figure 7: L
dependence on temperature
att
Integrating the fit-function over the temperature interval Ts− Tzallows us to extract the mean
value of the attenuation length:
• temperature range 80-124 K: < L
• temperature range 80-139 K: < L
>= 1 182 340 m ≈ 1.2 ·106m;
att
>= 936 521 m ≈ 9.4 · 105m.
att
7
Astrophysics in Antarctica Homework 01 October 7, 2023
The dielectric constant of water
Meissner, Thomas and Wentz, F.J. (2004). The Complex Dielectric Constant of Pure and Sea Water from Microwave Satellite Observations. IEEE Transactions on Geoscience and Remote Sensing. 42.
1836 - 1849.
Real and Imaginary part of the dielectric constant for pure water with a temperature of 0◦C
as function of frequency shown at Figure 8. The estimated value is ε
water
= 90 + 4i.
Figure 8: The dielectric constant of water at 0◦C as function of frequency
The final results
The amplitude of the transmitted signal (Eq. 7):
• Using first method of estimation the attenuation length
– Little America, Antarctica
5 ·1.38 · 10
−23m2
· kg · s−2· K−1· 80 K ·106Hz
VT=
10 ·10
8π ·50 · 103m
·
· e
1 m
– Iceland, the glacier ice
5 ·1.38 · 10
−23m2
· kg · s−2· K−1· 80 K ·106Hz
VT=
10 ·10
8π ·50 · 103m
·
1 m
(50·103m/28.7 m)
(50·103m/28.7 m)
· e
· 50Ω ·0.933
= 0.064 ·e
· 50Ω ·0.933
= 0.064 ·e
1742
308
!
1/2
V (!!!)
!
1/2
V (!!!)
·
·
• Using second method of estimation the attenuation length
5 ·1.38 · 10
−23m2
· kg · s−2· K−1· 80 K ·106Hz
VT=
10 ·10
8π ·50 · 103m
·
(50·103m/9.4·105m)
· e
1 m
8
· 50Ω ·0.933
= 0.067 V
!
1/2
·
Astrophysics in Antarctica Homework 01 October 7, 2023
The time delay of the sigal (Eq. 9):
∆t =
1.77
· 2 ·50 ·103m = 5.9 ·10−4s ≈ 0.6 ms.
3 ·108m/s
9
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