### Doppler impact on noticed break within the energy spectrum

Statement of magnetometer time sequence confirmed that break within the energy spectrum systematically happens at frequencies which can be greater than the gyrofrequency (Figs. 1 and three). Fluctuations in a shifting medium are described when it comes to oscillations as noticed within the body shifting with the medium modified by a Doppler shift.

$$start{aligned} omega ‘ = omega + textbf{okay} cdot textbf{v} finish{aligned}$$

(1)

In Jupiter’s magnetosphere Doppler shift impact could be written as.

$$start{aligned} textbf{okay} cdot textbf{v} sim k_perp v_perp finish{aligned}$$

(2)

The place (perp) implies amount perpendicular to the primary magnetic discipline. Right here (v_perp = v sin (theta _{vB}))^{13,19} and (theta _{vB}) is the angle between plasma bulk movement velocity and primary magnetic discipline in Jupiter solar orbit (JSO) coordinate system^{7}. Right here notice that measurement of (delta B) is a measurement of the amplitude of the oscillation and never wavelength or path of the wave propagation. This simplified estimate of the dot product assumes that waves detected by the spacecraft are propagating alongside the path of the perpendicular part of the majority velocity, shifting in direction of the spacecraft.

It’s cheap to imagine that the wavelength of the perpendicular fluctuation of the sector at ion gyrofrequency is corresponding to the circumference of ion gyration (k_perp (f_g)rho _i sim 1) (comparable cycles of movement of ions and magnetic discipline). On this work it’s deduced that Doppler shift is chargeable for the distinction between noticed frequency of the break and gyrofrequency (Fig. 1). Then Eq. (1) could be written when it comes to breakfrequency, gyrofrequency and gyroradius.

$$start{aligned} 2pi f_b sim 2pi f_g + frac{v_perp }{rho _i} finish{aligned}$$

(3)

The ion gyroradius is calculated when it comes to the ion temperature.

$$start{aligned} rho _i = frac{sqrt{m_ik_bT_i}}{eB} finish{aligned}$$

Introducing Doppler shift fixed

$$start{aligned} c_{ds} = frac{f_b – f_g}{f_g} finish{aligned}$$

equation (3) could be rewritten as

$$start{aligned} 2pi c_{ds} f_g sim frac{v_perp }{rho _i} finish{aligned}$$

Substituting for gyrofrequency and gyroradius it may possibly then be solved for the ion temperature

$$start{aligned} T_i sim frac{m_iv_perp ^2}{c_{ds}^2k_B} finish{aligned}$$

(4)

Notice that in Eq. (2) a bulk movement in Jupiter’s plasmadisc was assumed to be sooner than the wave velocity. In order that fluctuations of waves shifting in direction of the sensor are noticed. In additional common scenario, the Doppler shift (textbf{okay}cdot textbf{v}) can very nicely be detrimental. In that case break within the energy spectrum will likely be noticed at frequencies decrease than the gyrofrequency (f_b < f_g). In precept, magnitude of the angle between bulk velocity and wave propagation could be wherever between 0 and (pi). If path to the supply of the wave is thought then the extra common expression for the ion temperature is

$$start{aligned} T_i sim left( frac{hat{textbf{n}}_k cdot hat{textbf{n}}_v}{c_{ds}}proper) ^2frac{m_iv^2}{k_B} finish{aligned}$$

Because the angle between (textbf{okay}) and (textbf{v}) approaches (pi /2), (f_b) will method (f_g) in order that each (c_{ds}) and (hat{textbf{n}}_k cdot hat{textbf{n}}_v) will turn out to be small with canceling impact and because the angle between (textbf{okay}) and (textbf{v}) approaches (pi) each (c_{ds}) and (hat{textbf{n}}_k cdot hat{textbf{n}}_v) will turn out to be detrimental. Additionally notice that (c_{ds} > -1 forall f_b) and (f_g>0).

In absence of a greater estimate for (hat{textbf{n}}_k cdot hat{textbf{n}}_v) temperature was calculated utilizing Eq. (4) with instances within the vary (f_b> 2f_g). Right here bulk velocity modeled as a corotation velocity as much as (20, {textrm{R}_{textrm{J}}}), After that velocity is (textbf{v} = textbf{v}_phi = 200hat{textbf{e}}_phi {textrm{km/s}})^{18}. Close to the planet gyro and breakfrequencies for instances described above improve with improve within the power of the magnetic discipline. In samples from magnetometer measurements^{8} the place break is past the higher sure of the vary utilized by the algorithm (1.5times 10^{-1} {textrm{Hz}}) (Fig. 1b) or the place break in energy spectrum was not discovered, breakfrequency is estimated by the median worth (Fig. 3).

Determine 5 exhibits distribution of temperature estimates. Temperature calculations made with breakfrequency discovered by the subroutine are proven with blue markers. These estimates evaluate pretty nicely to the plasma temperature match from Galileo mission^{18} with the unfold that’s much like the unfold in reported temperature measurements. Temperature calculations made utilizing the median estimate for the breakfrequency are proven with purple markers. These estimates land on the Galileo profile. In order that if one wouldn’t hassle to attempt to discover a break within the energy spectrum and simply calculated it as (5f_g) then Eq. (4) will give the temperature profile from^{18}.

An intriguing chance is the usage of exterior fluctuations to set off cascade in shifting plasma nearer to gyrofrequency. On this case each (c_{ds}) and (v_b) in Eq. (4) will lower. This in precept might enable to transform kinetic vitality of the plasma movement into thermal. It’s potential that waves generated by magnetic exercise in Jupiter’s magnetosphere are used to precipitate a break within the energy spectrum nearer to gyrofrequency. The quantity of kinetic vitality transferred to thermal, then is dependent upon the distinction between gyro and breakfrequencies. In Jupiter’s magnetosphere this course of is inefficient, the unfold of the distinction between break and gyrofrequencies is about an order of magnitude (Fig. 3). This then ends in the pure unfold of temperature values^{18} and Fig. 5.

### Doppler impact on common frequency spectra

Assuming that the Doppler impact is comparable for fluctuations at totally different frequencies a extra common relation between noticed wave numbers and wave numbers within the bulk movement reference body could be made. In that case a relation between noticed and unique frequencies is much like the relation between break and gyro frequencies.

$$start{aligned} frac{f’-f}{f} sim frac{f_b-f_g}{f_g} = c_{ds} finish{aligned}$$

Right here the noticed frequency of the break within the energy spectrum is used as a marker for the Doppler shift of noticed wave fluctuations in the remainder of the frequency vary. So then Eq. (1) could be solved for *okay*.

$$start{aligned} okay sim left( frac{c_{ds}}{c_{ds}+1}proper) frac{2pi f’}{v (hat{textbf{n}}_k cdot hat{textbf{n}}_v)} finish{aligned}$$

(5)

Notice that Eq. (5) will deviate from Taylor’s speculation^{20} when noticed frequency is similar to the unique frequency and (c_{ds}ll 1), so within the case of a really low velocity or a really lengthy wavelength. It permits for the statement of waves with an arbitrary angle between instructions of wave and bulk velocities. In order that this permits an statement of incoming wave fluctuations from noticed frequency, bulk velocity and the breakfrequency within the energy spectrum. Alternatively, if wavenumber options *okay*(*f*) in relaxation body are identified then utilizing noticed (f’) and (f_b) Eq. (5) could be solved for path to the supply (hat{textbf{n}}_k cdot hat{textbf{n}}_v).