Gemechu Chala Debela is doing his Post Graduate studies at Wollega University, Ethiopia. He published his presentation here about Particle Emission. It will be useful to the Post Graduate and research scholars.

1. Power point Presentation on:
Particle Emission from the Sun and Fluid Flow in a Nozzle
By: Gemechu Chala Debela
Course name: Space Physics
Program: Post graduate(M.sc.)
Wollega University,Nekemte, Ethiopia
email: gamdebela@gmail.com
February 17, 2021

2. Outline of the Talk
1 Particle Emission from the Sun
Solar Wind
2 Fluid Flow In a Nozzle
Types of Nozzles
Flow Analyis
Gemechu Ch. (Wollega University)
Particle Emission from the Sun and Fluid Flow in a Nozzle
February 17, 2021 2 / 21

3. Particle Emission from the sun
Solar wind
A quiet Sun radiates not only electromagnetic waves, but also
particles.
This particle is called a solar wind, which is always blowing
and it varies quite strongly in velocity as well as density at a
distance from the Sun corresponding to the Earth’s orbit.
The solar wind is fully ionized, electrically neutral plasma that
carries a magnetic field and made of high-speed particles,
mostly protons and electrons, traveling rapidly outward from
the inner solar corona at all times.
Gemechu Ch. (Wollega University)
Particle Emission from the Sun and Fluid Flow in a Nozzle
February 17, 2021 3 / 21

4. Cont...
Under quiet conditions the wind has a medium velocity of 400
km/s but can, however, vary between 200 and 700 km/s
(Figure 1.1)
Figure: 1.1 Histogram of occurrence frequency for the values of solar
wind velocity, in interplanetary space. (From Hundhausen et al.,
1970.))
Note that with a speed (v) of 435 km/s the solar wind takes 4
days to reach the Earth from the Sun.
Gemechu Ch. (Wollega University)
Particle Emission from the Sun and Fluid Flow in a Nozzle
February 17, 2021 4 / 21

5. Cont...
The solar wind particle density (n) can vary between 1×106
and 2×107
m−3
.
The average energy of the protons is of the order of 1kev, while
the electrons have energies of the order of 1ev.
The average particle flux from the Sun can then be estimated
to be:
φ = n · v ≈ 2 × 1012
m−2
s−1
(1)
Total average particle loss per unit time from the Sun is
therefore.
Ṅ =
dN
dt
= 4πR2
Θ · φ ≈ 1.23 × 1031
s−1
(2)
Where: RΘ is the solar radius.
Gemechu Ch. (Wollega University)
Particle Emission from the Sun and Fluid Flow in a Nozzle
February 17, 2021 5 / 21

6. cont...
By neglecting the mass of the electrons in the solar wind, this
particle loss per unit time corresponds to a mass loss per unit
time given by:
ṁ =
dm
dt
=
dN
dt
· mp = 2.05 × 104
kg/s (3)
Where mp = 1.672 ×10−27
kg is the proton mass.
This can be compared with the mass loss per unit time due to
the blackbody radiation from the Sun and equal to 4.3 ×109
kg/s as derived below:
From Einstein’s energy-mass relation we have:
E = mc2
(4)
Gemechu Ch. (Wollega University)
Particle Emission from the Sun and Fluid Flow in a Nozzle
February 17, 2021 6 / 21

7. Cont...
The total radiated power (Q) from the Sun then corresponds
to a mass loss per unit time:
dm
dt
=
E
c2
=
Q
c2
=
σ · T4
· S
c2
(5)
Where: σT4
is the radiated power per unit area (q). σ is
Stephan-Boltzmann constant (5.67 × 10−8
Wm−2
K−4
) S is the
surface of the radiating body, and C is the speed of light
For a blackbody temperature of the Sun T = 5860K or close to
6,000K the radiated power per unit area will be:
q = σT4
= 5.67 × 10−8
Wm−2
K−4
× (5860 K)4
= 6.3×
107
Wm−2
(6)
Gemechu Ch. (Wollega University)
Particle Emission from the Sun and Fluid Flow in a Nozzle
February 17, 2021 7 / 21

8. Cont...
Since the solar radius is R = 7 × 108
m the total radiated
power from the Sun is
Q = 4πR2
· q = 4 × 3.14 × 7 × 108
m
2
× 6.3 × 107
Wm−2
= 3.9 ×1026
W(7)
Then the mass loss per unit time:
dm
dt
=
3.9 × 1026 W
(3 × 108)∧
2
= 4.3 × 109
kg/s (8)
The proton temperature in the solar wind is 1×104
- 2× 105
K
while the electron temperature is a factor of 3-4 larger during
quiet average conditions.
Gemechu Ch. (Wollega University)
Particle Emission from the Sun and Fluid Flow in a Nozzle
February 17, 2021 8 / 21

9. Cont...
During disturbed condition the proton and electron
temperatures are rather similar.
The magnetic field strength associated with solar wind is
varying between 1 and 15γ (1γ=10−9
tesla) (figure 1.2).
Figure: 1.2 Histogram of occurrence frequency for magnetic field strength
values in interplanetary space. (From Ness, 1969; e.g., Fa¨lthammar,
1973.))
Gemechu Ch. (Wollega University)
Particle Emission from the Sun and Fluid Flow in a Nozzle
February 17, 2021 9 / 21

10. Cont...
The characteristic parameters we have given for the solar wind
so far,we notice that the speed of sound of the solar wind gas is
approximately:
cs =
s
γ
kT

11. mp
= 1.17 × 104
m/s (9)
where:γ is the adiabatic constant equal to 5/3 for a
monoatomic gas and T

12. is set equal to 104
K.
This high-flow wind is produced by the pressure difference
between the hot dense gas of the solar atmosphere and the cold
tenuous gas in the background interstellar medium.
This pressure difference easily overcomes the solar
gravitational pull on the plasma.
Gemechu Ch. (Wollega University)
Particle Emission from the Sun and Fluid Flow in a Nozzle
February 17, 2021 10 / 21

13. Cont...
At a heliocentric distance rh the pressure in the solar wind
must balance the pressure of the interstellar medium, and the
solar wind must again be subsonic.
The change from supersonic plasma to subsonic plasma is
expected to be related to a shock wave, and at this shock the
interplanetary magnetic field terminates, forming the boundary
of the heliosphere, the heliopause.
It is at the moment impossible to determine the distance to this
shock, but it is believed to be situated between 60 and 100AU.
Gemechu Ch. (Wollega University)
Particle Emission from the Sun and Fluid Flow in a Nozzle
February 17, 2021 11 / 21

14. Fluid Flow In a Nozzle
Fluid Flow In a Nozzle
To get an idea of the mechanism behind the acceleration of the
solar wind, we will first discuss fluid flow through a nozzle.
Nozzle is a duct by flowing through which the velocity of a
fluid increases at the expense of pressure drop.
Nozzle is used in steam and gas turbines, in rocket motors, in
jet engines, and in many other applications.
Gemechu Ch. (Wollega University)
Particle Emission from the Sun and Fluid Flow in a Nozzle
February 17, 2021 12 / 21

15. Types of Nozzles
Types of Nozzles
Convergent nozzle: where the flow is subsonic
Figure: 1.3 Convergent nozzle
Convergent – Divergent nozzle: where the flow is supersonic
Gemechu Ch. (Wollega University)
Particle Emission from the Sun and Fluid Flow in a Nozzle
February 17, 2021 13 / 21

16. Flow Analyis
Flow Analyis
Compressible Flow: Compressible flow means a flow that
undergoes a notable variation in density with trending
pressure.
Incompressible flow:refers to the fluid flow in which the
fluid’s density is constant.
For a density to remain constant, the control volume has to be
remain constant.
Even though the pressure changes, the density will be constant
for an incompressible flow.
Let an incompressible gas with density ρ stream through a
tube with a varying cross-section A.
Gemechu Ch. (Wollega University)
Particle Emission from the Sun and Fluid Flow in a Nozzle
February 17, 2021 14 / 21

17. Cont...
Figure: 1.5 Mass flows through a nozzle with a minimum cross-section to
explain the presence of a critical region in the mass flow in order for the
flow speed to become supersonic.)
Since mass flux through any cross-section of the tube must be
constant, we have:
φm = A · ρ · v = const (10)
Where:v is the velocity of the gas through the cross-sectionA.
Gemechu Ch. (Wollega University)
Particle Emission from the Sun and Fluid Flow in a Nozzle
February 17, 2021 15 / 21

18. Cont...
Bernoulli’s equation is applicable only when flow is assumed to
be incompressible.
In case of compressible flow, Bernoulli’s equation becomes
invalid since the very basic assumption for Bernoulli’s equation
is densityρ is constant.
According to the equation of Bernoulli the pressure gradient
will be balanced by the inertia force when no other forces are
acting in the equation of motion.
Considering only one dimension along r:
Using Hydrostatic equilibrium in mathematical term.
dP
dr
= −ρg (11)
Gemechu Ch. (Wollega University)
Particle Emission from the Sun and Fluid Flow in a Nozzle
February 17, 2021 16 / 21

19. Cont...
dP
dr
= −ρg = −ρ
dv
dt
= −ρ
dv
dr
∗
dr
dt
= −ρ
dv
dr
· v (12)
And by dividing byρ and rearranging
dP
ρ
= −v · dv (13)
Rewriting dp by another partial derivative we have:
dP
ρ
=
dP
dρ
∗
dρ
ρ
= −v · dv (14)
And divide both side of equation (14) by dP
dρ
we get:
dρ
ρ
= −v · dv
dp
dρ
−1
(15)
Gemechu Ch. (Wollega University)
Particle Emission from the Sun and Fluid Flow in a Nozzle
February 17, 2021 17 / 21

20. Cont...
If we now assume that this process of the gas flow is adiabatic,
we have:
p · ρ−γ
= const. (16)
Where:γ is the adiabatic constant. Then by differentiating:
dp
dρ
= γ
p
ρ
= c2
s (17)
Where, Cs is the speed of sound in the gas.
inserting equation (17) into equation (15)
dρ
ρ
= −
v
c2
s
dv (18)
According to the conservation of mass flux (10) we can form:
dφm
φm
=
dA
A
+
dρ
ρ
+
dv
v
= 0 (19)
Gemechu Ch. (Wollega University)
Particle Emission from the Sun and Fluid Flow in a Nozzle
February 17, 2021 18 / 21

21. Cont...
By inserting for dρ
ρ
from (18) we get:
dA
A
−
v
c2
s
dv +
dv
v
= 0 (20)
Thus, the equation that relate the increase and decrease of
solar wind speed, the supersonic (v Cs) and relation of
cross-section area A of flow can be given as And
dA
A
=
v
c2
s
dv −
dv
v
=
v
c2
s
−
1
v
dv =
1
v
v2
c2
s
− 1
dv(21)
When v = Cs, dA
A
must be zero and the cross-section will not
decrease any more,and the flow is called Sonic flow
Gemechu Ch. (Wollega University)
Particle Emission from the Sun and Fluid Flow in a Nozzle
February 17, 2021 19 / 21

22. Cont...
As long as the cross-section A and the velocity increases (dA
A
;
negative and dv
v
; positive) the velocity v must be smaller than
Cs that is the flow is subsonic.
If, on the other hand, v is going to be larger than Cs, i.e.,
supersonic speed dA
A
must be positive and the cross-section
must increases again.
If, however, the gas is streaming so slowly that it does not
reach the speed of sound at the narrowest part of the tube the
speed of the gas will then continue to decrease as it passes
through the throttle.
Gemechu Ch. (Wollega University)
Particle Emission from the Sun and Fluid Flow in a Nozzle
February 17, 2021 20 / 21

23. Cont...
Thank You.
Gemechu Ch. (Wollega University)
Particle Emission from the Sun and Fluid Flow in a Nozzle
February 17, 2021 21 / 21