Solar-wind predictions for the Parker Solar Probe orbit
AAstronomy & Astrophysics manuscript no. paperMVVB c (cid:13)
ESO 2017November 22, 2017
Solar-wind predictions for the Parker Solar Probe orbit
Near-Sun extrapolations derived from an empirical solar-wind model based onHelios and OMNI observations
M. S. Venzmer and V. Bothmer
University of Goettingen, Institute for Astrophysics, Friedrich-Hund-Platz 1, 37077 Göttingen, GermanyReceived 25 August 2017; accepted 10 November 2017
ABSTRACT
Context.
The Parker Solar Probe (PSP) (formerly Solar Probe Plus) mission will be humanity’s first in situ exploration of the solarcorona with closest perihelia at 9 .
86 solar radii ( R (cid:12) ) distance to the Sun. It will help answer hitherto unresolved questions on theheating of the solar corona and the source and acceleration of the solar wind and solar energetic particles. The scope of this study isto model the solar-wind environment for PSP’s unprecedented distances in its prime mission phase during the years 2018–2025. Thestudy is performed within the Coronagraphic German And US SolarProbePlus Survey (CGAUSS) which is the German contributionto the PSP mission as part of the Wide-field Imager for Solar PRobe (WISPR). Aims.
We present an empirical solar-wind model for the inner heliosphere which is derived from OMNI and Helios data. The German-US space probes Helios 1 and Helios 2 flew in the 1970s and observed solar wind in the ecliptic within heliocentric distances of0 . .
98 au. The OMNI database consists of multi-spacecraft intercalibrated in situ data obtained near 1 au over more than fivesolar cycles. The international sunspot number (SSN) and its predictions are used to derive dependencies of the major solar-windparameters on solar activity and to forecast their properties for the PSP mission.
Methods.
The frequency distributions for the solar-wind key parameters, magnetic field strength, proton velocity, density, and temper-ature, are represented by lognormal functions. In addition, we consider the velocity distribution’s bi-componental shape, consistingof a slower and a faster part. Functional relations to solar activity are compiled with use of the OMNI data by correlating and fittingthe frequency distributions with the SSN. Further, based on the combined data set from both Helios probes, the parameters’ frequencydistributions are fitted with respect to solar distance to obtain power law dependencies. Thus an empirical solar-wind model for theinner heliosphere confined to the ecliptic region is derived, accounting for solar activity and for solar distance through adequate shiftsof the lognormal distributions. Finally, the inclusion of SSN predictions and the extrapolation down to PSP’s perihelion region enablesus to estimate the solar-wind environment for PSP’s planned trajectory during its mission duration.
Results.
The CGAUSS empirical solar-wind model for PSP yields dependencies on solar activity and solar distance for the solar-windparameters’ frequency distributions. The estimated solar-wind median values for PSP’s first perihelion in 2018 at a solar distanceof 0 .
16 au are 87 nT, 340 km s − , 214 cm − and 503 000 K. The estimates for PSP’s first closest perihelion, occurring in 2024 at0 .
046 au (9 . R (cid:12) ), are 943 nT, 290 km s − , 2951 cm − , and 1 930 000 K. Since the modeled velocity and temperature values belowapproximately 20 R (cid:12) appear overestimated in comparison with existing observations, this suggests that PSP will directly measuresolar-wind acceleration and heating processes below 20 R (cid:12) as planned. Key words. solar wind – sun: heliosphere – sun: corona
1. Introduction
From observations of cometary tail fluctuations, Biermann(1951) inferred the presence of a continuous flow of particlesfrom the Sun. With his theoretical solar-wind model, Parker(1958) formulated the existence of the solar wind even beforethe first satellites measured it in situ in 1959 (Gringauz et al.1960; Neugebauer & Snyder 1966). The idea of a space missionflying through the solar corona dates back to the founding yearof NASA in 1958 (McComas et al. 2008). Since then severalspace missions have measured the solar wind in situ at a widerange of heliocentric distances. In the case of Voyager 1, thiswas as far away as 140 au in October 2017, having crossed theheliospause into interstellar space at a distance of 121 au (Gur-nett et al. 2013). Various spacecraft have provided a wealth ofsolar-wind measurements near Earth’s orbit, with WIND (Lep-ping et al. 1995; Ogilvie et al. 1995), SOHO (Domingo et al. https://voyager.jpl.nasa.gov/ . .
31 au. Helios 2, launched twoyears later, approached the Sun as close as 0 .
29 au (Rosenbaueret al. 1977). The NASA Parker Solar Probe (PSP), formerlySolar Probe Plus, six years after its planned launch date in mid2018, will reach its closest perihelia at a distance of 9.86 solarradii ( R (cid:12) ), that is, 0 . http://parkersolarprobe.jhuapl.edu/ Article number, page 1 of 14 a r X i v : . [ a s t r o - ph . S R ] N ov & A proofs: manuscript no. paperMVVB be achieved through seven Venus gravity assists with orbital pe-riods of 88–168 days. In its prime mission time 2018–2025 PSPprovides 24 orbits with perihelia inside 0 .
25 au (Fox et al. 2015).Even its first perihelion, 93 days after launch in 2018, will takePSP to an unprecedented distance of 0 .
16 au (35 . R (cid:12) ). In com-parison, the ESA Solar Orbiter mission with a planned launch inFebruary 2019 will have its closest perihelia at 0 .
28 au (Mülleret al. 2013).The key PSP science objectives are to “trace the flow of en-ergy that heats and accelerates the solar corona and solar wind,determine the structure and dynamics of the plasma and mag-netic fields at the sources of the solar wind, and explore mecha-nisms that accelerate and transport energetic particles” as statedin Fox et al. (2015). To achieve these goals, PSP has four sci-entific instruments on board: FIELDS for the measurement ofmagnetic fields and AC / DC electric fields (Bale et al. 2016),SWEAP for the measurement of flux of electrons, protons andalphas (Kasper et al. 2016), IS (cid:12)
IS for the measurement of solarenergetic particles (SEPs) (McComas et al. 2016) and WISPRfor the measurement of coronal and inner heliospheric structures(Vourlidas et al. 2016).The study presented in this paper is undertaken in the Coro-nagraphic German And US SolarProbePlus Survey (CGAUSS)project, which is the German contribution to the PSP mission aspart of the Wide-field Imager for Solar PRobe (WISPR). WISPRwill contribute to the PSP science goals by deriving the three-dimensional structure of the solar corona through which the insitu measurements are made to determine the sources of the solarwind. It will provide density power spectra over a wide range ofstructures (e.g., streamers, pseudostreamers and equatorial coro-nal holes) for determining the roles of turbulence, waves, andpressure-balanced structures in the solar wind. It will also mea-sure the physical properties, such as speed and density jumps ofSEP-producing shocks and their coronal mass ejection (CME)drivers as they evolve in the corona and inner heliosphere (Vourl-idas et al. 2016). In order to help optimize the WISPR andPSP preplanning of the science operations, knowledge of the ex-pected solar-wind environment is needed. For this purpose thesolar-wind environment is extrapolated down to the closest peri-helion of 9 . R (cid:12) distance to the Sun using in situ solar-wind datafrom the Helios probes and near 1 au data from various satellitescompiled in the OMNI solar-wind database.Generally, two types of solar wind are observed in theheliosphere – slow and fast streams (Neugebauer & Snyder1966; Schwenn 1983). Slow solar wind has typical speedsof <
400 km s − and fast solar wind has speeds >
600 km s − (Schwenn 1990, p. 144). Their di ff erent compositions and char-acteristics indicate di ff erent sources and generation processes(McGregor et al. 2011b). Fast streams are found to originatefrom coronal holes as confirmed by Ulysses’ out-of-eclipticmeasurements (McComas et al. 1998). The source of slow wind,and its eventually di ff erent types (Schwenn 1983), is still a sub-ject of controversial discussions because several scenarios arepossible to explain its origin from closed magnetic structures inthe solar corona, such as intermittent reconnection at the top ofhelmet streamers and from coronal hole boundaries (Kilpua et al.2016). The occurrence frequency of these slow and fast streamsvaries strongly with solar activity and their interactions lead tophenomena such as stream interaction regions which may per-sist for many solar rotations ("co-rotating" interaction regions)if the coronal source regions are quasi-stationary (Balogh et al.1999). Embedded in the slow and fast solar-wind streams aretransient flows of CMEs – the faster ones driving shock wavesahead (Gosling et al. 1974). Their rate follows the solar activity cycle and varies in near 1 au measurements between only oneCME every couple of days during solar cycle minima up to mul-tiple CMEs observed over several days at times of solar maxima,that is, the CME-associated flow share of the solar wind raisesfrom about 5 % up to about 50 % (Richardson & Cane 2012).It is not known which specific solar-wind type or structurePSP will encounter at a given time during its mission, there-fore we extrapolate the probability distributions of the majorsolar-wind parameters from existing solar-wind measurementsand take solar cycle dependencies into account. As a baselinewe describe the solar-wind environment through the key quanti-ties of a magnetized plasma: magnetic field strength, density andtemperature. Furthermore, the bulk flow velocity is the definingparameter of the two types of solar wind. Solar-wind quantities,like flux densities, mass flux, and plasma beta, can be directlyderived from these four parameters. In the analyses, we treat thesolar wind as a proton plasma – the average helium abundanceis about 4 .
2. Frequency distributions of the solar-windparameters
The solar-wind parameters are highly variable due to short-termvariations from structures such as slow and fast wind streams, in-teraction regions, and CMEs, whose rate and properties dependon the phase of the solar activity cycle. Hence, for deriving char-acteristic frequency distributions for the solar-wind parameters,measurements over long-term time spans are needed. The abun-dance of the near-Earth hourly OMNI data set is ideally suitedfor this purpose, because to date it spans almost five solar cycles.The OMNI 2 data set (King & Papitashvili 2005) combinessolar-wind magnetic field and plasma data collected by varioussatellites since 1963, currently by WIND and by ACE. This in-tercalibrated multi-spacecraft data is time-shifted to the nose ofthe Earth’s bow shock. The data is obtained from the OMNI-Web interface at NASA’s Space Physics Data Facility (SPDF),Goddard Space Flight Center (GSFC). In this study the wholehourly data until 31 December 2016 is used, starting from 27November 1963 (for the temperature from 26 July 1965). Thedata coverage of the di ff erent parameters is in the range 67–74 %, corresponding to a total duration of 36–40 years. We notethat a test-comparison of hourly averaged data with higher-time-resolution data for the available shorter time span 1981–2016 didnot show significant di ff erences in our results. According to the http://omniweb.gsfc.nasa.gov/ Article number, page 2 of 14. S. Venzmer and V. Bothmer: Solar-wind predictions for the Parker Solar Probe orbit
OMNI data precision and maximal parameter ranges we spec-ify bin sizes of 0 . − for the velocity, 1 cm − for the density and 10 000 K for thetemperature. The frequency distributions of the solar-wind mag-netic field strength, proton velocity, density and temperature areshown in Fig. 1. The solar-wind magnetic field strength is in therange 0 . − , thedensity in the range 0–117 cm − , the temperature in the range3450–6 . × K, and the mean data values are at 6 .
28 nT,436 km s − , 6 . − and 1 . × K. These ranges and meanvalues are as statistically expected from previous analyses ofnear 1 au solar-wind data (e.g., Table 3.3 in Bothmer & Daglis(2007, p. 39)). Much higher or lower peak values at 1 au havebeen observed in extraordinary events, such as the 23 July 2012CME with a speed of over 2000 km s − and a peak field strengthof about 100 nT that was observed by STEREO A (Russell et al.2013), or the solar-wind disappearance event observed in May1999 with density values even down to 0 . − (Lazarus 2000).The frequency distributions of the solar-wind parameters,magnetic field strength, proton density, and temperature, canbe well approximated by lognormal distributions, whereas theproton velocity’s frequency has a di ff ering shape, as shown inVeselovsky et al. (2010). We investigate how well all four solar-wind parameters’ frequency distributions can be represented bylognormal functions, which we use in the process of a leastsquares regression fitting. The lognormal function, W ( x ) = σ √ π x exp (cid:32) − (ln x − µ ) σ (cid:33) , (1)depends on the location µ and the shape parameter σ . Changes in µ a ff ect both the horizontal and vertical scaling of the functionwhereas σ influences its shape. The distribution’s median x med and mean x avg (average) positions are easily interpreted and aredirectly calculated from µ and σ : x med = exp ( µ ) ⇐⇒ µ = ln ( x med ) , (2) x avg = exp (cid:32) µ + σ (cid:33) ⇐⇒ σ = (cid:115) (cid:32) x avg x med (cid:33) . (3)It is apparent that the mean is always larger than the median. Re-placing the variables µ and σ with these relations, the lognormalfunction (1) becomes W ( x ) = (cid:113) π ln (cid:16) x avg x med (cid:17) x exp − ln (cid:16) xx med (cid:17) (cid:16) x avg x med (cid:17) . (4)The values of x med and x avg obtained from fitting the individualsolar-wind frequency distributions are listed in Table 1.From visual inspection, the resulting fit curves describe theshape of the magnetic field strength, the density and the temper-ature distributions well, as can be seen in Fig. 1. However, forthe velocity, the fit function appears not to be as good in describ-ing the measured distribution’s more complex shape around itspeak and in the higher velocity range. This also can be inferredfrom the sum of absolute residuals (SAR) between data and fit,listed in Table 1 as a percentage of the distribution area, beingalmost three times larger than those from the other parameters.In order to find a better fit result for the velocity distribution, weassume that the velocity distribution can be made up of at leasttwo overlapping branches (McGregor et al. 2011a). Therefore a compositional approach is chosen by combining two lognormalfunctions (4), involving more fit variables: W II ( x ) = c · W ( x ) + (1 − c ) · W ( x ) . (5)The balancing parameter c ensures that the resulting function re-mains normalized as it represents a probability distribution. Thefitting of W II ( x ) to the velocity’s frequency distribution yields thevalues of the now five fit parameters ( c , x med,1 , x avg,1 , x med,2 and x avg,2 ) as listed in Table 1 together with the median and meanvalues of the composed distribution, which can be derived bysolving (cid:90) W II ( x ) d x = (cid:90) x W II ( x ) d x = . (6)This more complex fit function is more accurate in describingthe velocity’s frequency distribution as shown in Fig. 2. Thus inthe following Sections we keep the double lognormal ansatz forall velocity frequency fits.For the bulk of the solar wind these static lognormal func-tions describe the parameters’ distributions well. The abnor-mally high parameter values in the distribution functions canbe attributed to shock / CME events in agreement with the re-sults of the OMNI solar-wind investigations by Richardson &Cane (2012). The simple lognormal fit functions underestimatethe frequencies in their high-value tails, except for the tempera-ture’s tail which is overestimated, as seen in the insets of Fig. 1.This appears to be because CMEs do not come with abnormallyhigh temperatures, but rather with temperatures lower than thoseof the average solar wind (Forsyth et al. 2006). The velocity’scompositional lognormal fit only slightly overestimates its tailas seen in the inset of Fig. 2. The slow and fast part contributealmost equally ( c ≈ .
5) to the long-term velocity distributionfunction.
3. Solar activity dependence of the solar-windfrequency distributions
In the next step we investigate how the long-term solar-winddistribution functions presented in the previous section dependon general solar activity. Therefore we examine their correlationwith the SSN, being a commonly used long-term solar activityindex, and determine the time lags with the highest correlationcoe ffi cients.For the correlations we fit lognormal functions to the fre-quency distributions as in Sect. 2, but implement linear relationsto the yearly SSN, allowing shifting of the distribution func-tions with SSN. For the velocity the approach is di ff erent in-sofar as its two components are kept fixed and instead their bal-ance is modified with the changing SSN. Thus we obtain solar-activity-dependent models for the frequency distributions of allfour solar-wind parameters.The international sunspot number (1963–2016) is providedby the online catalog at the World Data Center – Sunspot Indexand Long-term Solar Observations (WDC-SILSO), Solar Influ-ences Data Analysis Center (SIDC), Royal Observatory of Bel-gium (ROB).Yearly medians of the solar-wind parameters and the yearlySSN together with the solar cycle number are shown in the up-per part of Fig. 3. The reason for correlating the SSN to thesolar-wind median values is because the position of a lognor-mal function is defined by its median. The data are averaged to Article number, page 3 of 14 & A proofs: manuscript no. paperMVVB
Table 1.
Resulting fit coe ffi cients from the fitting of the lognormal function (4) to the shape of the solar-wind parameters’ frequency distributionsfrom near 1 au OMNI hourly data. For the velocity, the fit parameters of the double lognormal function (5) are also listed, as well as the medianand mean values of the resulting velocity fit. The numbers in parentheses are the errors on the corresponding last digits of the quoted value. Theyare calculated from the estimated standard deviations of the fit parameters. For each parameter, the sum of absolute residuals between data and fit(in percentage of the distribution area) is also listed. Parameter Median Mean Balance SAR x med x avg c [%]Magnetic field [nT] 5 . . . km s − ] 4 . . . − ] 5 . . . K] 7 . . . W . . km s − ] W . . W II . a . a – 4 . Notes. ( a ) Error estimates derived from the individual fit part errors. F r equen cy Magnetic field strength [nT] OMNI dataLognormal fitFit medianFit mean 0 0.02 0.04 0.06 0.08 0.1 1 10 F r equen cy Magnetic field strength [nT] 0 0.01 0.02 0.03 0.04 0.05 0 200 400 600 800 1000 F r equen cy Velocity [km s -1 ] 0 0.01 0.02 0.03 0.04 0.05 0 200 400 600 800 1000 F r equen cy Velocity [km s -1 ] 0 0.05 0.1 0.15 1 10 100 F r equen cy Density [cm -3 ] 0 0.05 0.1 0.15 1 10 100 F r equen cy Density [cm -3 ] 0 0.02 0.04 0.06 0.08 0.1 10 F r equen cy Temperature [K] 0 0.02 0.04 0.06 0.08 0.1 F r equen cy Temperature [K]
Fig. 1.
Frequency distributions of the four solar-wind parameters and their lognormal fits derived from the hourly OMNI data set. The histogramshave bins of 0 . − , 1 cm − and 10 000 K. The fits’ median and mean values are indicated as well. The insets show zoomed-in views ofthe high-value tails of the distributions. yearly values to avoid seasonal e ff ects during the Earth’s orbitaround the Sun caused by its variations in solar latitude and dis-tance. The solar-wind velocity, density, and temperature dependon the state of the solar cycle (Schwenn 1983). For instance thefast solar wind occurs at times when polar coronal holes extendto lower latitudes, a typical feature of the declining phase of thesolar cycle as pointed out by Bothmer & Daglis (2007, p. 75,Fig. 3.52). Therefore the solar-wind velocity, density, and tem-perature maxima exhibit time lags relative to the SSN maxima. The correlation coe ffi cients of the solar wind parameterswith the yearly SSN shown in the bottom part of Fig. 3 are calcu-lated for time lags back to −
15 years to cover a time span longerthan a solar cycle. As expected, the amplitudes of the variationsin the correlations of all parameters decline with increasing timelag and show a period of about 11 years. The highest correlationcoe ffi cient of 0.728 to the SSN is found for the magnetic fieldstrength; it has no time lag. This finding is anticipated becausethe SSN is found to be directly proportional to the evolution ofthe photospheric magnetic flux (Smith & Balogh 2003). Veloc- Article number, page 4 of 14. S. Venzmer and V. Bothmer: Solar-wind predictions for the Parker Solar Probe orbit F r equen cy Velocity [km s -1 ] OMNI dataFit partsComposite fitFit medianFit mean 0 0.01 0.02 0.03 0.04 0.05 0 200 400 600 800 1000 F r equen cy Velocity [km s -1 ] Fig. 2.
Velocity frequency distribution (same as in Fig. 1) and its com-positional lognormal fit. The fit’s median and mean values and its twofit parts are indicated as well. The inset is a zoomed-in view of the highvalue tail of the distribution. M agne t i c f i e l d [ n T ]
200 300 400 500 600 V e l o c i t y [ k m s - ] D en s i t y [ c m - ] T e m pe r a t u r e [ K ] SS N -0.8-0.4 0 0.4 0.8 -14 -12 -10 -8 -6 -4 -2 0Correlation of solar wind parameters to lagged SSN C o rr e l a t i on c oe ff i c i en t Time lag [years]
Magnetic fieldVelocityDensityTemperature
Fig. 3.
Solar-wind parameter yearly frequencies (gray shading) withyearly medians (lines) derived from OMNI data and the yearly SSNfrom the SILSO World Data Center (1963–2016) with solar cycle num-ber (top). Their correlation coe ffi cients with the yearly SSN are calcu-lated for time lags back to -15 years (bottom). ity and temperature show time lags of 3 years with peak corre-lation coe ffi cients of 0.453 and 0.540. The density with a corre-lation coe ffi cient of 0.468 has a time lag of 6 years, which is inagreement with the density anticorrelation to the SSN reportedby Bougeret et al. (1984).Next we create solar-activity-dependent analytical represen-tations of the solar wind frequency distributions. This is achieved by shifting the median positions of the lognormal distributionsas a linear function of the SSN. To enable these shifts, we add alinear SSN dependency to the median, x med ( ssn ) = a med · ssn + b med , (7)using a factor to the SSN a med with a baseline b med . We relatethe mean with a scaling factor to the median to transfer its SSNdependency: x avg ( ssn ) = (cid:16) + a avg (cid:17) · x med ( ssn ) . (8)These relations, substituted into the lognormal function (4), leadto a new SSN-dependent function W (cid:48) ( x , ssn ). This function isthen fitted to the yearly data, using the yearly SSN as inputparameter. The SSN is o ff set with the individual time lags de-termined before for each parameter, to benefit from the highercorrelation. The values of the three resulting fit coe ffi cients( a med , b med and a avg ) are presented in Table 2.Naturally, the fit models match with the general data trends,as can be seen from Fig. 4, though single year variations are notreplicated by the model (e.g., the high velocity and temperaturevalues in 1974, 1994, and 2003). The comparison of this modelwith the yearly data median values with respect to the laggedSSN shows that the medians obtained from the modeling have asimilar slope, as shown in Fig. 5.Again, the solar-wind velocity needs a special treatment be-cause of the application of the double lognormal distribution (5).Since it is well known that slow and fast solar-wind stream oc-currence rates follow the solar cycle, we keep the two velocitycomponents’ positions SSN-independent ( x med = b med ) and varyinstead their balance with the SSN: c ( ssn ) = c a · ssn + c b . (9)The fit result (see Table 2) yields a model in which three yearsafter solar cycle minimum (SSN of zero) the contribution of slowsolar wind to the overall solar wind distribution reaches a max-imum value (about 64 %) and decreases with increasing SSN asshown in Fig. 6.To investigate the amount of slow and fast wind contribu-tions depending on solar activity, we apply the commonly usedconstant velocity threshold of v th =
400 km s − (Schwenn 1990,p. 144). The linear fit to the yearly data ratio and the derivedmodel ratio show a good agreement (see Fig. 6). The to-some-degree steeper balance parameter of the double fit function usedin this model cannot be compared directly with specific velocitythresholds between slow and fast solar wind. However, it appearsto be a more realistic approach than just taking a specific veloc-ity threshold for the slow and fast wind, in agreement with theoverlapping nature of the velocity flows reported by McGregoret al. (2011a).
4. Solar distance dependency
In order to derive heliocentric distance relationships of the bulksolar wind distribution functions, we apply and fit power lawdependencies to the Helios data. We then examine how the fitsmay be extrapolated towards the Sun and in particular in to thePSP orbit. We use the fitting methods of Sect. 2 for the distance-binned combined data from both Helios probes. Helios’ highlyelliptical orbits in the ecliptic covered a solar distance range of0 . .
98 au in case of Helios 1 and 0 . .
98 au in case of He-lios 2. Launched during solar cycle minimum, the data of both
Article number, page 5 of 14 & A proofs: manuscript no. paperMVVB
Table 2.
Resulting fit coe ffi cients from the OMNI data, based on the linear SSN dependencies (7) and (8). For the velocity the fit parameters fromthe double lognormal fit (5) and their balancing function (9) are given. The numbers in parentheses are the errors on the corresponding last digitsof the quoted value. They are calculated from the estimated standard deviations of the fit parameters. The listed SSN time lags are used for the fits. Parameter Median Mean Balance SSN lagSSN factor a med Baseline b med Scaling factor a avg SSN factor c a Baseline c b [years]Magnetic field [nT] 1 . × − . . × − – – 0Density [cm − ] 3 . × − . . × − – – 6Temperature [10 K] 1 . × − . . × − – – 3Velocity W (cid:48) – 3 . . × − − . × − km s − ] W (cid:48) – 4 . . × − − . × − M agne t i c f i e l d [ n T ] -4 -3 -2 -1 F r equen cy DataData
ModelModel SS N Actual SSNNo shift D en s i t y [ c m - ] -4 -3 -2 -1 F r equen cy DataData
ModelModel SS N Actual SSN6-year shift V e l o c i t y [ k m s - ] -4 -3 -2 -1 F r equen cy DataData
ModelModel SS N Actual SSN3-year shift T e m pe r a t u r e [ K ] -4 -3 -2 -1 F r equen cy DataData
ModelModel SS N Actual SSN3-year shift
Fig. 4.
Solar wind parameter yearly data frequencies and lognormal fit models, both with their median values (white lines) over the OMNI timeperiod 1963–2016. The corresponding yearly SSN and the shifted SSN for the models are indicated by gray and black lines. The velocity medianis derived from the SSN-weighted constant lognormal parts (dotted lines). probes cover the rise to the maximum of cycle 21, covering ∼ ∼ ∼ / SPDF .The Helios 1 magnetometer data coverage for this data setis about 43 % (i.e., 2.8 years), and that of Helios 2 amounts to54 % (i.e., 2.3 years). The plasma data coverage is 76 % (i.e.,5.0 years) in case of Helios 1 and 92 % (i.e., 3.9 years) in case http://spdf.gsfc.nasa.gov/ of Helios 2. Thus, using this data, we point out that its time cov-erage is unequally distributed over the solar cycle. Consideringthe data gap distributions, the amount of data during solar cycleminimum up to mid 1977, that is, the transition from minimumto maximum, covers about 68 % of this period whereas duringmaximum of cycle 21 data are available only 38 % of the time.This Helios data bias towards solar minimum is one reason whyin this study the Helios solar wind data are not used to derivelong-term frequency distributions and solar-cycle dependenciesfor the key solar wind parameters.The radial dependencies of the key solar-wind parametersover the distance range 0 . .
98 au measured by both Heliosprobes are plotted in Fig. 7, together with their median and meanvalues for di ff erent solar distances, calculated for the minimaldistance resolution 0 .
01 au of the data set. Assuming a radial
Article number, page 6 of 14. S. Venzmer and V. Bothmer: Solar-wind predictions for the Parker Solar Probe orbit M agne t i c f i e l d [ n T ] Yearly medianData linear fitModel median
300 350 400 450 500 550 600 0 50 100 150 200 250 V e l o c i t y [ k m s - ] D en s i t y [ c m - ] SSN 5101520 0 0 50 100 150 200 250 T e m pe r a t u r e [ K ] SSN
Fig. 5.
Solar-wind parameter medians with respect to the lagged SSN.The yearly data medians ( + ) with their weighted linear fit (solid lines)are obtained from OMNI data. The error bars denote the SSN stan-dard deviation and the relative weight from the yearly data coverage.The SSN-dependent median (dashed lines) is derived from the lognor-mal model fit. For the velocity the median is derived from the SSN-weighting (9) of the slow and fast model parts (dotted lines), whosemagnitudes are SSN independent. B a l an c e o f s l o w t o f a s t w i nd SSN
Yearly ratio ( v th = 400 km s -1 )Data linear fit Model balance parameter c Model ratio ( v th = 400 km s -1 ) Fig. 6.
Ratio of slow to fast solar wind for a SSN lagged by 3 years. Theyearly ratios ( + ) and their weighted linear fit (solid line) are obtainedfrom OMNI data with a threshold velocity of v th =
400 km s − . Theerror bars denote the SSN standard deviation and the relative weightfrom the yearly data coverage. The model’s balance parameter (9) andderived ratio (same threshold) are plotted as dashed and dotted lines. solar-wind outflow, it is expected that the distance dependenceof the solar-wind parameters over the Helios data range 0 . .
98 au can be described through power law scaling. Thereforewe use the power law function, x ( r ) = d · r e , (10)for the regression fit of the median and mean, with r being thesolar distance in astronomical units, d the magnitude at 1 au and e the exponent. The fits are weighted through the di ff erent datacounts per bin. The obtained coe ffi cients for the median andmean power law fits ( d med , e med , d avg and e avg ) are listed in Ta-ble 3 and their corresponding curves are shown in Fig. 7.Our derived exponents agree with those found in existingstudies from the Helios observations: Mariani et al. (1978) de-rived the exponents for the magnetic field strength separately forthe fast and slow solar wind as B fast ∝ r − . and B slow ∝ r − . ,ours is B avg ∝ r − . . The velocity exponent v avg ∝ r . matcheswith the values found by Schwenn (1983, 1990), who derived the distance dependencies for both Helios spacecraft separatelyas v H1 ∝ r . and v H2 ∝ r . . The calculated density exponent n avg ∝ r − . agrees well with the Helios plasma density modelderived by Bougeret et al. (1984) yielding n ∝ r − . . The tem-perature exponent T avg ∝ r − . is similar to those in the studiesby Hellinger et al. (2011, 2013), who also derived the exponentsseparately for the fast and the slow solar wind: T fast ∝ r − . and T slow ∝ r − . .The mean and median velocity fit exponents acquired fromthe Helios data are very similar, which indicates that they canbe kept identical so that the basic shape of the frequency distri-bution does not change with distance. Conversely, the mean andmedian fits for the magnetic field strength cross each other at0 .
339 au (see Table 3) and the mean becomes slightly lower thanthe median at smaller distances. Thus, below that distance thefrequency distribution can no longer be well described by a log-normal function, because the mean of a lognormal function hasto be larger than its median (as pointed out in Sect. 2), that is,the location of the crossing indicates that the parameter’s distri-bution is no longer of a lognormal shape thereafter. The fits forthe proton temperature show a similar behavior, having an ex-trapolated intersection at 0 .
082 au. Therefore the extrapolationof the magnetic field and temperature distribution frequencies tothe PSP orbit by applying lognormal functions is limited. Thecrossing points limit the regions where the distribution’s shapescan still be considered lognormal.In order to still fit and extrapolate lognormal functions withthe data, we assume that the shapes can be considered lognormalat all distances. For the frequency distribution fit function to bediscussed in the following paragraph, we reduce the fit exponents e med and e avg to only one. We note that this simplification leads toslightly larger modeling errors, especially in case of the magneticfield strength.Next we retrieve the frequency distributions of the four so-lar wind parameters in solar distance bins of 0 .
01 au, choosingthe same resolution as for the OMNI data analyzed in Sect. 2 –the distributions and their median values are plotted in Fig. 8.For simplification, as mentioned before, we treat the exponentsof the median and mean fit functions as being identical, usingone fit parameter for both. Implementing the power law distancedependency (10) into the lognormal function (4), we get the fitparameters d (cid:48) med , d (cid:48) avg and the common exponent e (cid:48) . Again, weuse the double lognormal function (5) for the velocity distribu-tion fit – resulting in W (cid:48)(cid:48) II ( x , r ). The additional fit parameters arethe balancing parameter c (cid:48) and for the second lognormal part d (cid:48) med,2 and d (cid:48) avg,2 . The resulting fit coe ffi cients for the four solarwind parameters are presented in Table 4.The velocity balancing parameter c (cid:48) = .
557 is in goodagreement with the results for the SSN dependency (9), be-cause with a mean SSN of 59 during the Helios time period, c (59) = .
53, as can be seen from Fig. 6.The power law lognormal models and the power law dou-ble lognormal model for the velocity, which result from the fit-ting, are plotted in Fig. 8 together with their median values.The model’s magnetic field strength is broader around values of40 nT at the lower distance boundary than the data’s frequencydistribution implies. This behavior is expected because of theapplied distance-independent shape approximation. The velocityand temperature models’ upper values generally show a higherabundance than the actual data; see also zoom boxes in Figs. 1and 2. The high-velocity tail that increases with distance arisesfrom using the same exponent for both slow and fast compo-nents. This e ff ect is not seen in the data; more specifically, not Article number, page 7 of 14 & A proofs: manuscript no. paperMVVB M agne t i c f i e l d s t r eng t h [ n T ] Helios dataMeanMedianMean fitMedian fit V e l o c i t y [ k m s - ] CMEs > 800 km s -1 D en s i t y [ c m - ] Solar distance [au] 1 10 100 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 T e m pe r a t u r e [ K ] Solar distance [au]10 Fig. 7.
Helios hourly data plots of the four solar wind parameters over solar distance. The mean and median per 0 .
01 au data bin and their fit curvesare plotted as well. The Helios data has a native distance resolution of 0 .
01 au, thus, to make the distribution visible in these plots, we added arandom distance value of up to ± .
005 au. The high velocity data points above 800 km s − (circled red) are identified as CME events (e.g., Sheeleyet al. 1985; Bothmer & Schwenn 1996, 1998). Table 3.
Fit coe ffi cients for the median and mean solar distance dependencies (10) of the four solar wind parameters derived from the combinedHelios 1 and 2 data. The numbers in parentheses are the errors on the corresponding last digits of the quoted value. They are calculated from theestimated standard deviations of the fit parameters. The crossing distances indicate where the median and mean fits intersect each other. The yearlyvariation is the weighted standard deviation derived from the yearly fit exponents seen in Fig. 9. Parameter Median Mean Crossing distance Yearly variation d med e med d avg e avg [au] ∆ e Magnetic field [nT] 5 . − . . − . . . km s − ] 4 . . . . . × . − ] 5 . − . . − . . . K] 7 . − . . − . . . Table 4.
Fit coe ffi cients for the distance-dependent single lognormal function, based on Equation (4) combined with (10) from the combined Heliosdata. Regarding the velocity, the double lognormal function (5) is used instead. The numbers in parentheses are the errors on the corresponding lastdigits of the quoted value. They are calculated from the estimated standard deviations of the fit parameters. The seasonal variations are calculatedfrom Earth’s orbital solar distance variation and the derived exponents. Parameter Median Mean Exponent Balance Seasonal variation d (cid:48) med d (cid:48) avg e (cid:48) c (cid:48) ∆ d [%]Magnetic field [nT] 5 . . − . . − ] 5 . . − . . K] 6 . . − . . W (cid:48)(cid:48) . . . . . km s − ] W (cid:48)(cid:48) . . . . . W (cid:48)(cid:48) II . a . a – – – Notes. ( a ) Velocity median and mean 1 au values for the resulting function. Error estimates derived from the individual fit part errors.Article number, page 8 of 14. S. Venzmer and V. Bothmer: Solar-wind predictions for the Parker Solar Probe orbit M agne t i c f i e l d s t r eng t h [ n T ] DataData -4 -3 -2 -1 F r equen cy ModelModel D en s i t y [ c m - ] DataData -4 -3 -2 -1 F r equen cy ModelModel V e l o c i t y [ k m s - ] DataData -4 -3 -2 -1 F r equen cy ModelModel T e m pe r a t u r e [ K ] DataData -4 -3 -2 -1 F r equen cy ModelModel
Fig. 8.
Frequency distributions of the four solar wind parameters with respect to solar distance. Plotted are the binned Helios data and the powerlaw lognormal fit models with their median values (white lines). The double lognormal model is used for the velocity, its slow and fast parts areindicated by dotted lines. only the slowest wind but also the fastest wind is expected toconverge to more average speeds (Sanchez-Diaz et al. 2016).
5. Empirical solar-wind model
In order to estimate the solar-wind environment for the PSP or-bit, we combine the results from the solar-wind frequency dis-tributions’ solar-activity relationships and their distance depen-dencies derived from the OMNI and Helios data. The result isan empirical solar-wind model for the inner heliosphere whichis then extrapolated to the PSP orbit in Sect. 6.This solar-wind model for the radial distance dependence isrepresentative for the time of the Helios observations around therise of solar cycle 21. The variations of the yearly power lawfit exponents from fitting the solar-distance dependency (10) areshown in Fig. 9 together with the yearly SSN for the time period1974–1982. It can be seen that during the Helios time periodthere might be some systematic variation of the exponents withsolar activity – at least for the velocity and temperature expo-nents. However, for simplicity we assume that the distance scal-ing laws can be treated as time independent and include the cal-culated exponents’ yearly variations ∆ e , summarized in Table 3,as relative uncertainties.Since we neglect possible variations of the distance scalinglaws, we combine the frequency distribution’s median solar ac-tivity dependency (7) derived for 1 au from the OMNI data with -3-2.5-2-1.5-1-0.5 0 0.5 1 1.5 1974 1975 1976 1977 1978 1979 1980 1981 1982 0 50 100 150 200 250 E x ponen t SS N SIDC smoothed SSNMagnetic fieldVelocityDensityTemperature
Fig. 9.
Helios yearly variation of the solar wind parameter power ex-ponents for the dependence on radial distance together with the SIDC13-month smoothed monthly SSN. The weighted standard deviationsand average values for all years are indicated by the shaded areas. Inthis plot, the 21 days since Helios launch in the year 1974 are omittedbecause a distance range of merely 0 . .
98 au was covered that year. the power law exponents (10) derived from the Helios data: x med ( ssn , r ) = ( a med · ssn + b med ) · r e (cid:48) . (11) Article number, page 9 of 14 & A proofs: manuscript no. paperMVVB
Thus, implementing the median and mean relations into the log-normal function (4), we obtain the combined model function W (cid:48)(cid:48)(cid:48) ( x , ssn , r ) and for the velocity W (cid:48)(cid:48)(cid:48) II ( x , ssn , r ) with the dou-ble lognormal function (5). The corresponding median and meanrelations for each solar-wind parameter, based on the values re-sulting from our analyses, are listed below. Their numerical val-ues are the fit parameters from Table 2 and the exponents fromTable 4. – The magnetic field strength relations, depending on solar ac-tivity and solar distance, are: B med ( ssn , r ) = (0 . · ssn + .
29 nT) · r − . , (12) B avg ( ssn , r ) = . · B med ( ssn , r ) . (13) – The proton velocity relations for the slow and fast compo-nents, depending on solar distance, are: v slowmed ( r ) =
363 km s − · r . , v fastmed ( r ) =
483 km s − · r . , (14) v slowavg ( r ) = . · v slowmed ( r ) , v fastavg ( r ) = . · v fastmed ( r ) . (15)The share of both components balanced with solar activity isfound to be: c ( ssn ) = − . · ssn + . . (16) – The derived relations of the proton density are: n med ( ssn , r ) = (cid:16) . − · ssn + .
50 cm − (cid:17) · r − . , (17) n avg ( ssn , r ) = . · n med ( ssn , r ) . (18) – The derived proton temperature relations are: T med ( ssn , r ) = (197 K · ssn +
57 300 K) · r − . , (19) T avg ( ssn , r ) = . · T med ( ssn , r ) . (20)These relations average over seasonal variations becausethey are based on yearly data. The OMNI data are time-shiftedto the nose of the Earth’s bow shock; this leads to yearly solardistance variations of ± .
67 % as it orbits the Sun. The result-ing maximal solar-wind parameter variation amplitudes over theyear can thus be derived from the derived power law exponents.They are estimated to be smaller than 4 % as seen in Table 4.Bruno et al. (1986) and Balogh et al. (1999) have pointed outthat the solar-wind parameters vary with latitudinal separationfrom the heliospheric current sheet. Its position in heliographiclatitude is highly variable around the solar equator (Schwenn1990) and, furthermore, the Earth’s orbit varies over the courseof the year by ± . ◦ in latitude. Since this latitudinal separationis highly variable and requires significant e ff ort to calculate foran extended time series, we have ignored this aspect in this anal-ysis.
6. Model extrapolation to PSP orbit
To estimate PSP’s solar-wind environment during its missiontime for its orbital positions, predictions of the SSN during themission are incorporated into the empirical solar-wind model,derived in the previous Sections, and extrapolations down to thePSP perihelion region are performed.Parker Solar Probe is planned to launch in mid 2018. With itsfirst Venus flyby it will swing into Venus’ orbital plane, reach-ing a first perihelion with a distance of 0 .
16 au just 93 days after S o l a r d i s t an c e [ au ] PSP orbitVenus flybysExaminedperihelia SS N SIDC smoothed actualSIDC predictionSWPC predictionCycle 24 shifted by 11 yearsHalf/twice cycle 24 amplitude SS N Fig. 10.
PSP’s solar distance during its mission time (top). ConsecutiveVenus flybys bring its perihelia nearer to the Sun. Actual and predictedSSN (bottom), that is, SIDC 13-month smoothed monthly actual SSN,SIDC Standard Curves Kalman filter prediction and SWPC predictionwith their corresponding expected ranges (shaded areas). The SSN fromprevious cycle 24, shifted by 11 years, is plotted together with two al-ternative trends of half and twice its amplitude. launch, in November 2018. Seven additional Venus flybys allowthe perihelion distance to be reduced to a minimum of 9 . R (cid:12) .This distance will be reached with the 22nd perihelion in Decem-ber 2024 (Fox et al. 2015), as plotted in the top panel of Fig. 10.We extrapolate the derived empirical solar-wind models toPSP’s orbital distance range and compare the results with thosefrom the existing models shown in Fig. 11. The model and its ex-trapolation are visualized for the SSN range between solar min-imum and maximum (0 ≤ ssn ≤ .
25 au to 715 nTat 0 .
046 au for a SSN of zero. Taking a SSN of 200 increasesthe values to 69 nT and 1152 nT. Our extrapolation results areslightly flatter than those derived from the analytical magneticfield model by Banaszkiewicz et al. (1998), who constructedan analytic dipole plus quadrupole plus current sheet (DQCS)model for solar minimum. We note that one cannot easily com-pare the absolute values of our study with the values obtained byBanaszkiewicz et al. (1998) because the DQCS model assumessolar wind originating from coronal holes at higher heliographiclatitudes only, neglecting the slow solar-wind belt. We suggestthat the di ff erence in slope is due to the previously mentioned(Sect. 4) changing shape of the frequency distribution with he-liocentric distance, which for smaller distances deviates morefrom the model’s lognormal distribution. The average velocity isfound to decrease from 340 km s − at 0 .
25 au to about 290 km s − at 0 .
046 au 3 years after a SSN of zero occurred, whereas us-ing a SSN of 200 it decreases from 390 km s − to 330 km s − .Comparing the results with the measurements by Sheeley et al.(1997) and Wang et al. (2000) shows an overestimation in ourextrapolated slow solar-wind velocity values for distances belowapproximately 20 R (cid:12) . They used LASCO coronagraph observa-tions to track moving coronal features (blobs) in the distancerange 2–30 R (cid:12) to determine speed profiles and sources of theslow solar wind and they derived temperature and sonic pointvalues for slow solar wind with the isothermal expansion modelfrom Parker (1958). Therefore, it generally can be expected that Article number, page 10 of 14. S. Venzmer and V. Bothmer: Solar-wind predictions for the Parker Solar Probe orbit M agne t i c f i e l d s t r eng t h [ n T ] [ R ☉ ] Model, 0 ≤ ssn ≤ ≤ ssn ≤ V e l o c i t y [ k m s - ] [ R ☉ ] Slow and fast partsParker solution, 1.1x10 K, 5 R ☉ (Sheeley et al. 1997)Parker solution, 10 K, 5.8 R ☉ (Wang et al. 2000)HeliosPSP HeliosPSP D en s i t y [ c m - ] Solar distance [au][ R ☉ ] Leblanc et al. (1998), 5 % HeHeliosPSP T e m pe r a t u r e [ K ] Solar distance [au][ R ☉ ] Billings et al. (1959),Liebenberg et al. (1975) HeliosPSP HeliosPSP
Fig. 11.
Radial extrapolation of the solar-wind parameters to the PSP orbit region. The median values from the models, obtained from Helios andOMNI measurements, are extrapolated to the PSP region for SSN values between solar minimum and maximum, that is, 0 ≤ ssn ≤ PSP will encounter a slower solar-wind environment close tothe Sun than our model estimates. Thus PSP will measure solar-wind acceleration processes (McComas et al. 2008), maybe evenstill at 30 R (cid:12) as the study by Sheeley et al. (1997) suggests. Theproton density increases from about 84 cm − at 0 .
25 au to about3018 cm − at 0 .
046 au 6 years after a SSN of zero occurred. Be-ing almost independent of the SSN, the values for a SSN of 200are only 17 % larger. The results are in good agreement withthose of Leblanc et al. (1998), who derived an electron den-sity model from type III radio burst observations. Their modelshows that the density distance dependency scales with r − andsteepens just below 10 R (cid:12) with r − . We assumed a solar-windhelium abundance of 5 % to convert these electron densities toproton densities. The extrapolated proton temperature increasesfrom about 260 000 K at 0 .
25 au to about 1 690 000 K at 0 .
046 au3 years after a SSN of zero occurred and from 440 000 K to2 860 000 K for a SSN of 200. Knowing that near-Sun coronaltemperatures are in the range of 2–3 MK (Billings 1959; Lieben-berg et al. 1975), the model overestimates the extrapolated tem-peratures at the PSP perihelion distance.Aside from the solar distance, the derived solar-wind pa-rameter models depend on the SSN. Short-term predictions ofthe SSN can be used for the solar-wind predictions of PSP’searly perihelia and also for refining the solar-wind predictionsduring PSP’s mission. Several sources are available for SSN short-term predictions. The SIDC provides 12-month SSN fore-casts obtained from di ff erent methods (e.g., Kalman filter Stan-dard Curve method). The SSN prediction of NOAA’s SpaceWeather Prediction Center (SWPC) for the time period until theend of 2019 follows a consensus of the Solar Cycle 24 Predic-tion Panel . The SSN for PSP’s first perihelion will be small –certainly below 20 – whereas the SSN during the closest per-ihelia, which will commence at the end of 2024 at the likelymaximum phase of cycle 25, cannot be predicted at this time.However, Hathaway & Upton (2016) found indications that thenext solar cycle will be similar in size to the current cycle 24.Therefore we simply assume a pattern similar to the last cyclefor the prediction of the next solar cycle and thus shift the lastcycle by 11 years. Additionally, we consider as possible alterna-tives SSN patterns of half and twice its amplitude as shown inthe bottom panel of Fig. 10.Implementing the SSN predictions for the PSP mission timeand the orbital trajectory data, we can infer which solar-windparameter magnitudes can be expected. Figures 12 and 13 showthe median values (7) of the considered di ff erent solar-wind pa-rameters for 12-day periods, comprising the first perihelion in http://sidc.be/silso/forecasts Article number, page 11 of 14 & A proofs: manuscript no. paperMVVB M agne t i c f i e l d [ n T ]
250 300 350 400 450 50024.10. 28.10. 01.11. 05.11. V e l o c i t y [ k m s - ] D en s i t y [ c m - ] T e m pe r a t u r e [ K ] Fig. 12.
Estimated solar-wind parameter medians (black lines) and theirerror bands (gray) during 12 days in 2018 with PSP’s first perihelion atabout 0 .
16 au. For the velocity the combined median is calculated andalso the SSN-independent slow and fast parts are plotted (dotted lines). M agne t i c f i e l d [ n T ]
250 300 350 400 450 50012.12. 16.12. 20.12. 24.12. V e l o c i t y [ k m s - ] D en s i t y [ c m - ] T e m pe r a t u r e [ K ] Fig. 13.
Estimated solar-wind parameter medians (black lines) and theirerror bands (gray) during 12 days in 2024 with PSP’s 22nd (the firstclosest) perihelion at 0 . Novemver 2018 and the first closest perihelion in December2024. In the beginning of the mission median values of about87 nT, 340 km s − , 214 cm − and 503 000 K are estimated to bemeasured at 0 .
16 au, increasing to about 943 nT, 290 km s − ,2951 cm − and 1 930 000 K during the first closest approach at0 .
046 au. Monthly SSNs – shifted by the time lags specific to thesolar-wind parameters – are used in the calculation of the solar-wind predictions. These SSNs are either actual smoothed valuesfrom the SIDC with their reported standard deviations, short-term predictions from the SWPC with their expected ranges, oractual smoothed values from the SIDC shifted by 11 years withhalf / twice their values as uncertainties. The error bands given inboth Figures, calculated from error propagation, include theseSSN ranges and the derived fit parameter errors.Finally the estimated solar-wind environment can be derivedfrom the function W (cid:48)(cid:48)(cid:48) ( x , ssn , r ). The estimated frequency distri-butions of the four solar-wind parameters at PSP’s 1st and 22nd F r equen cy Magnetic field [nT] 0 0.01 0.02 0.03 0.04 0.050 200 400 600 800 1000 F r equen cy Velocity [km s -1 ] OMNI dataOMNI fit1st perih.22nd perih. F r equen cy Density [cm -3 ] 0 0.02 0.04 0.06 0.08 0.1 10 F r equen cy Temperature [K]
Fig. 14.
Frequency distributions of the four solar-wind parameters(same as in Figs. 1 and 2) and those estimated with the solar-wind modelfor PSP’s 1st and 22nd (first closest) perihelion. In these Figures the fre-quencies of both extrapolated curves are scaled for visibility to the sameheight as the 1 au distribution. (first closest) perihelion are plotted in Fig. 14. Again, we pointout that the velocity and temperature distributions for the 22ndperihelion are only upper limits and the actual values to be en-countered by PSP are expected to be smaller.
7. Discussion and summary
The scientific objective of this study, being part of the CGAUSSproject – the German contribution to the WISPR instrument –is to model the solar-wind environment for the PSP mission tobe launched mid 2018. For this purpose we started the devel-opment of the empirical solar-wind environment model for thenear-ecliptic PSP orbit. We derived lognormal representationsof the in situ near-Earth solar-wind data collected in the OMNIdatabase, using the frequency distributions of the key solar-windparameters, magnetic field strength, proton velocity, density, andtemperature. Throughout the di ff erent analyses in our study, thevelocity’s frequency distribution is treated as a composition ofa slow and a fast wind distribution. Each velocity part is fittedwith a lognormal function, which allows for the overlap of bothvelocity ranges. The OMNI multi-spacecraft solar-wind data isintercalibrated and covers almost five solar cycles. It thus repre-sents solar wind gathered at di ff erent phases of solar activity inthe ecliptic plane. In the next step we investigated the yearly vari-ation of the solar-wind distribution functions along with the SSNover 53 years and derived linear dependencies of the solar-windparameters with the SSN. The radial dependencies of the solar-wind distribution functions were then analyzed, using Helios 1and 2 data for the distance range 0 . .
98 au in bins of 0 .
01 au,deriving power law fit functions that were used to scale the previ-ously calculated SSN-dependent 1 au distribution fit functions tothe PSP orbit, taking into account SSN predictions for the years2018–2025, encompassing the prime mission up to the closestapproach of 9 . R (cid:12) . The reason for performing the analysis thisway is based on the fact that the OMNI solar-wind database ismuch larger than the Helios database.For determining solar-activity- and solar-distance-dependentrelations for the median and mean solar-wind values, we couldhave used the simpler approach of combining the radial depen- Article number, page 12 of 14. S. Venzmer and V. Bothmer: Solar-wind predictions for the Parker Solar Probe orbit dence of averaged Helios data with averaged 1 au OMNI datascaled with the SSN. It is expected that the results of a simpleranalysis would have similar distance scaling results, as can beinferred from the exponents in Tables 3 and 4. However, in ourstudy we are not only interested in averages but rather in bulkdistributions, that is, the whole range of values that might occur.For the determination of the frequency distributions the use ofthe more complex fit model is important, because the distancebetween median and mean values determines the width of thelognormal distributions.It is clear that the calculated distribution functions only rep-resent first-order estimates of the real solar wind to be encoun-tered by PSP. The solar-wind environment to be encountered willdepend at times of PSP on the structure of the solar corona andunderlying photospheric magnetic field and on the evolution andinteraction of individual solar-wind streams and superimposedCMEs and shocks. However, the derived results are in goodagreement with existing studies about near-Sun solar-wind mag-netic field strengths and densities as shown in Sect. 6. The ex-trapolation results of the velocity and the temperature di ff er fromthe direct measurements seen in existing studies. This suggeststhat below about 20 R (cid:12) PSP may dive into the region where theacceleration and heating of the solar wind is expected to occur(see Fig. 11). The near-Sun solar-wind velocity at PSP perihelionis also expected to be slower than our model estimates, becausethe solar wind is assumed to be accelerated up to the height ofthe Alfvénic critical surface, which is predicted to lie on aver-age around 17 R (cid:12) (e.g., Sittler & Guhathakurta 1999; Exarhos& Moussas 2000), scaling with solar activity within a range ofbetween 15 R (cid:12) at solar minimum and 30 R (cid:12) at solar maximum(Katsikas et al. 2010; Goelzer et al. 2014).We have not specifically investigated the occurrences of ex-treme solar-wind parameters caused by CMEs or enhanced val-ues due to stream interaction or co-rotating interaction regions.The Helios solar-wind measurements plotted over radial distancein Fig. 7 show several extreme values far above the usual solar-wind velocities, which are associated with individual CMEs. Theresults by Sachdeva et al. (2017) indicate that due to solar-winddrag, the speeds of fast CMEs will commonly slow down sub-stantially from early distances of a few solar radii. Therefore,it is expected that PSP will encounter CMEs with much higherspeeds than those observed during the Helios mission. Also, themagnetic field, density and temperature values are expected tobe much larger than in the average solar wind in individual fast-shock-associated CME events. PSP will thus also substantiallyimprove our understanding of the near-Sun evolution of CMEsand their expansion with radial distance.With the resulting CGAUSS empirical solar-wind model forPSP, the following main results for the bulk solar-wind parame-ters and estimations for their median values at PSP’s first peri-helion in 2018 at a solar distance of 0 .
16 au and at PSP’s closestperihelia beginning in 2024 at 0 .
046 au (9 . R (cid:12) ) are obtained: – The dependency of the magnetic field strength on solar activ-ity and radial distance appears to be valid above 20 R (cid:12) , how-ever near PSP’s closest perihelia, the actual values might befound to be slightly higher. – The estimated magnetic field strength median values ob-tained from relation (12) for PSP’s 1st and 22nd perihelionare 87 nT and 943 nT. – The radial dependencies of the proton velocity median val-ues for slow and fast solar wind (14) appear to be validabove about 20 R (cid:12) solar distance; below they overestimatethe actual solar wind velocities obtained from remote mea-surements. The share of their frequency distributions to the overall solar-wind velocity distribution (5) depends on solaractivity with their balance relation (16). Thus, at solar mini-mum, with a SSN of around zero, the slow-wind componentcontributes about 64 % and drops to 28 % during solar max-imum conditions with a SSN around 200. – The calculated median velocity values for PSP’s 1st and22nd perihelion are 340 km s − and 290 km s − . – The proton density relation appears to be valid throughoutthe full PSP orbital distance range, even down to about 8 R (cid:12) . – The estimated density median values obtained from relation(17) for PSP’s 1st and 22nd perihelion are 214 cm − and2951 cm − . – The derived correlation function for the proton temperatureappears to provide overly high temperature values aroundPSP’s closest perihelion in comparison to coronal measure-ments. – The estimated temperature median values obtained from re-lation (19) for PSP’s 1st and 22nd perihelion are 503 000 Kand 1 930 000 K.The results of the modeled solar-wind environment will beuseful to help optimize the WISPR and in situ instrument scienceplannings and PSP mission operations. This also applies for theHeliospheric Imager (SoloHI) (Howard et al. 2013) and the insitu instruments on board the Solar Orbiter spacecraft.
Acknowledgements.
The authors acknowledge support of the CoronagraphicGerman and US SolarProbePlus Survey (CGAUSS) project for WISPR by theGerman Aerospace Center (DLR) under grant 50 OL 1601 as national con-tribution to the Parker Solar Probe mission. The authors thank the Helios andOMNI PIs / teams for creating and making available the solar wind in situ data.The Helios and the OMNI data are supplied by the NASA Space Science DataCoordinated Archive and the Space Physics Data Facility at NASA’s GoddardSpace Flight Center. Additional thanks for maintaining and providing the inter-national sunspot number series goes to the World Data Center – Sunspot Indexand Long-term Solar Observations at the Solar Influences Data Analysis Center,Royal Observatory of Belgium. The PSP SPICE kernel was kindly provided byAngelos Vourlidas. The authors thank the referee for the careful review of thismanuscript and helpful comments and suggestions. References
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