3.12 Summary
4.1.3 The length of the Parker spiral
In the Parker model, the HMF forms Archimedean spirals which lie on constant heliolatitude.
The length of the spiral magnetic field line is important since a charged particle cannot be detected at some arbitrary point in the heliosphere (e.g. between the Sun and the Earth) before it has travelled along the magnetic field line connecting those two points. To calculate the length of the Parker spiral, the following expression is used
S = Z r
r0
|d~l|, (4.14)
where the inner boundaryr0 is chosen to be0.05AU (in this study), and the outer boundary r is chosen to be 3 AU since the transport model is solved up to 3 AU. The integrand, d~l, is the infinite amount of small, incremental increases in the magnetic field line in spherical coordinates such that
(|d~l|)2 =dr2+r2dθ2+r2sin2θdφ2 (4.15) where, for the Parker HMFdφ=−Ωdr/Vsw, anddθ= 0, such that equation 4.15 becomes
52 4.1. INTRODUCTION Parker Spiral Lengths
Outer Boundary
Vsw = 400 km s−1
Vsw = 600 km s−1
Vsw = 800 km s−1
Vsw = 1000 km s−1
Vsw = 1200 km s−1 r1= 1AU 1.12AU 1.03AU 1.00AU 0.98AU 0.97AU r2= 2AU 3.02AU 2.50AU 2.29AU 2.17AU 2.11AU r3= 3AU 5.88AU 4.56AU 3.96AU 3.64AU 3.45AU
Table 4.1: Parker spiral magnetic field line length for different solar wind speeds. The inner boundary isr0 = 0.05AU.
|d~l|= s
dr2+r2sin2θΩ2dr2 Vsw2
= s
1 +r2sin2θ Ω2 Vsw2 dr.
(4.16)
By the definition of the winding angle (equation 4.4), equation 4.16 reduces to
|d~l|=p
1 + tan2ψ dr. (4.17)
Then, according to equation 4.14 the length of one magnetic field line is
S= Z r
r0
p1 + tan2ψ dr. (4.18)
From equation 4.18 it is clear that the length of the magnetic field line is directly dependent on the parameters encapsulated by the winding angle. Since the model is solved only in the equatorial plane, sinθ = 1 and Ω is constant (Ω = 25.4 days or 2.86×10−6 rad/sec). The assumed solar wind speed in the equatorial plane is400km s−1 (2.6×10−8 AU/s), although faster (or slower) solar wind speeds could be present in this plane. Accordingly, it is insightful to calculate the Parker spiral length for different solar wind speeds. Table 4.1 shows the magnetic field line length (in AU) for different solar wind speeds. The inner boundary isr0 = 0.05(the assumed radial distance to the SWSS).
Table 4.1 shows that for increasingly faster solar wind speeds, the length of the magnetic field lines decreases. In the extreme scenario of an almost straight magnetic field line from the Sun to the Earth, caused by a very fast solar wind speed, the field line cannot be shorter than the measured radial distance between the Sun and the Earth. Therefore, a limit on equation 4.18 appears. Note that since the inner boundary was chosen to ber0 = 0.05AU, the minimum magnetic field line length between the Sun and the Earth is0.95AU and this is confirmed by the top row of values (r1 = 1AU) from Table 4.1. Figure 4.1 shows the Parker magnetic field line length for different solar wind speeds of200,400,600, and800km s−1 as indicated in the legend. The shaded area represents the impossible scenario of an HMF line being shorter than the distance between the Sun and a chosen observation point a radial distanceraway. For easy reference, the Sun is indicated atr= 0AU, the Earth at1AU and Mars at1.5AU. The different HMF lines have very similar line lengths close to the Sun (r = 0AU andr = 0.3AU), after
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 Radial Distance (AU)
0 1 2 3 4 5 6 7
Parker spiral length (AU)
200 km s 1 400 km s 1 600 km s 1 800 km s 1 Sun Earth (1 AU) Mars (1.5 AU)
Figure 4.1: Parker spiral length for different solar wind speeds. The shaded area represents the impossible scenario of an HMF line being shorter than the distance concerned. The locations of the Sun, Earth, and Mars is indicated at0,1, and1.5AU, respectively.
which they diverge and the slower solar wind speeds have longer HMF lines and the faster solar wind speeds have shorter HMF lines.
Calculating the minimum amount of travel time of a particle (e.g. between the Sun and the Earth) could be useful to ensure a particle is not observed prematurely in the transport model.
To do this, particle speed as well as particle rigidity are calculated from special relativity argu- ments. Particle rigidity is determined by the following expression
Rp = q
Ep(Ep+E0), (4.19)
where Ep andE0 refer to the particle’s kinetic energy and the particle’s rest mass energy, re- spectively. From this, the ratio of particle speed to the speed of light is calculated as
βp = vp
c = Rp
Ep+E0. (4.20)
Figure 4.2 shows the particle speed for both electrons (dashed black line) and protons (dotted blue line) for particle kinetic energies from1keV to10GeV. For easy reference, the speed of light (7.2AU/hr) is indicated as a horizontal asymptote (dot-dashed red line) and the vertical green dot-dashed line indicates 85 keV electrons. The much later rise in proton speed is observed where∼200MeV protons have the same speed of the85keV electrons.
54 4.1. INTRODUCTION
10 3 10 2 10 1 10 0 10 1 10 2 10 3 10 4 Kinetic energy (MeV)
0 1 2 3 4 5 6 7
Particle speed (AU/hr)
Electrons Protons
Speed of light (7.2 AU/hr) 85 keV
Figure 4.2: Particle speeds for both electrons (dashed black) and protons (dotted blue) in the energy range from 1 keV to 10 GeV. The speed of light and the 85 keV particle energy are indicated by a horizontal dot-dashed red line and a vertical dot-dashed green line, respectively.
10
310
210
110
010
110
210
310
4Kinetic energy (MeV)
0 25 50 75 100 125 150 175 200
Minimum particle travel time to 1 AU (Min)
Electrons Protons
200 km s
1400 km s
1600 km s
1800 km s
18 min 85 keV
Figure 4.3: The minimum travel time of both electrons and protons along a magnetic field line from the Sun to the Earth (1AU) for different solar wind speeds. The minimum travel time between the Sun and the Earth at the speed of light is indicated by the horizontal dashed red line and the85keV electrons are shown by the vertical green dashed line to guide the eye.
From the particle speeds (figure 4.2) and the length of the magnetic field lines (figure 4.1), the minimum travelling time from the inner boundary at0.05AU outward along the Parker HMF is calculated for different solar wind speeds and shown in figure 4.3. The85keV electrons are indicated by a vertical dashed green line to guide the eye. The minimum amount of time for a particle to travel between the Sun and the Earth at the speed of light is∼ 8minutes (or7.2 AU/hr), shown as a horizontal red dashed line and serves as an asymptote for higher energy particles.