T f

The force by solar radiation acting on a dust particles is given by:

Here, a is the particle radius, p the dust particle density, Lq the solar luminosity, c the speed of light, and Qpr the radiation pressure efficiency given by the ratio of the radiation pressure cross-section to the geometrical cross section (Qpr ~2 for large particles).

Radiation and gravitational forces act in opposite directions and are both proportional to 1/r^. Thus, the dust particles move on Keplerian orbits around the Sun with what is effectively a gravitation field reduced by (1—3):

where the ¡-parameter is defined as the ratio of radiation pressure and gravitation force:

The effective acceleration of dust particles therefore depends on their size a. Generally, large dust particles remain near the nucleus for longer times, because their acceleration is low. Small particles are accelerated more efficiently. If the particles are very small and become transparent, their acceleration decreases again. Investigations of the spatial distribution of dust particles in the coma and tail can therefore provide indications on the dust particle size distribution. However, the radiation pressure efficiency, Qpr, depends on the composition of the dust and varies with its absorption and scattering properties. Therefore, it is difficult to disentangle size and material properties of dust particles from observations (see Sect. 7). Figure 34 shows the ¡-parameter for various materials as a function of grain radius for illustration.

The dynamics of dust particles on a large scale is determined by 3 and can be described by the formalism of [83]. They used the concept of synchrones and syndynes to describe the distribution of dust particles in the large-scale dust tail. Particles with any ¡-value ejected at the same time, t, from the nucleus are distributed along a line called "synchrone" (Fig. 35). Particles with the same ¡-value ejected at any time are distributed along "syndynes". For a given observing geometry, we can compute a set of synchrones and syndynes, with free parameters such as the particle size distribution and the initial velocity. The parameter set best fitting the appearance of the observed dust tail is used to derive the dust particle parameters. Usually, the optical properties and densities of dust particles are assumed to be the same, and only the size distribution is varied. Alternatively, Monte-Carlo models have been made to compute the motion of the dust particles. The inverse Monte-Carlo approach described by [87] takes into account, for example, a distribution of initial velocities and anisotropic outgassing. Reference [88] gives an overview

ig grain radiu-s [|jm] Fig. 34. The ^-parameter as a function of grain radius for different materials
Fig. 35. The principle of synchrone and syndyne calculations as performed by Finson and Probstein [65]

on the various methods to derive information on the dust particles from studies of their distribution in the dust tail.

Sometimes, a sunward spike of the dust tail is seen when the Earth crosses the orbital plane of a comet. This phenomenon is called an "anti-tail." The anti-tail is often interpreted as "old" and large particles with small ^-value, released months before the observations. Another feature sometimes observed during orbital plane crossing is called a "neck-line." Neck-lines appear as narrow, bright spikes in the dust tail aligned with the solar and anti-solar direction. As all particles ejected by a comet finally move along Keplerian orbits around the Sun, they cross the orbital plane again on the second node, 180° away from their ejection node. When the Earth passes through the orbital plane of the comet, these particles are seen lined-up in projection as a narrow spike and become bright by strong forward scattering. Possibly, most anti-tails reported in the past were actually unrecognized neck-line observations. Neck-lines are usually observed post-perihelion and are formed by dust particles ejected during the pre-perihelion path. Periodic comets could in principle also show neck-lines pre-perihelion, but this has never been observed. The interest in neck-lines arises from the potential to detect very large particles that are otherwise difficult to investigate in dust tail observations [88].

3.4 Dynamics of Ion Tails

Comet — Solar Wind Interaction

The parent and daughter molecules in the cometary coma are eventually ionized by photoionization, charge exchange or collisional ionization in the inner coma. The charged cometary particles interact with the solar wind. The solar wind consists mainly of hydrogen and helium ions. They stream approximately radially outward from the Sun with velocities of a few hundred kilometers per second, depending on heliographic latitude, solar cycle and interaction regions within the solar wind flow. The solar wind carries with it the magnetic field, which is "frozen" into the flow. This means the solar magnetic field lines are fixed to a fluid element and move outward with this flow element. Because the Sun rotates, the magnetic field lines therefore form a spiral around the Sun, the so-called Parker spiral [173]. The ionized cometary atoms and molecules form an obstacle in the solar wind, leading to a large interaction zone and the formation of a several 107 km long cometary ion tail.

For the following description of the comet - solar wind interaction, we choose the cometocentric frame of reference. In this frame, the solar wind streams at the comet. For simplicity, we assume the frozen magnetic field at an angle of 90° to the flow field, which streams straight at the comet (see Fig. 36). Figure 37 shows the physical parameters along the path of the Giotto spacecraft through the coma of comet Halley computed by a

magneto-hydrodynamical model. We use these figures to guide us through the main interaction zones of an active comet with the solar wind:

• As the solar wind approaches the comet, its magnetic field picks up an increasing number of cometary ions. This mass loading leads to a reduction of the solar wind speed with decreasing distance to the comet. Eventually, a bow shock forms in front of the comet separating the supersonic solar wind flow from the subsonic ion flow around the nucleus. In Fig. 37, a jump in velocity, temperatures, and magnetic field around 106 km marks the location of the bow shock.

• Behind the shock, the solar wind flow is increasingly mass loaded by cometary ions and the velocity further reduced. At large distances, sideways from the comet, however, the solar wind passes undisturbed. The interplanetary magnetic field, which is frozen into the solar wind flow, therefore folds (Fig. 38) around the comet [5].

• At the pressure boundary of the outstreaming cometary ions with the onstreaming mass loaded solar wind, an ionopause forms. Inside the ionopause, a magnetic field-free cavity forms (around 3.5 x 103 km in Fig. 37). Here, we find purely cometary plasma.

• In front of the ionopause, the magnetic field piles-up and the magnetic field strength increases.

• The temperatures of ions and electrons also drop from solar wind values to relatively cool conditions in the inner coma (Figs. 37 and 39).

• Magnetic curvature and pressure forces quickly accelerate the cometary ions up to velocities of a few hundred kilometers per second. The cometary ion tail forms.

log R

Fig. 37. Results of a 3D-MHD model for the ion tail of comet Halley during Giotto encounter [210]. Ion, Ti, and electron temperatures, Te, magnetic field, B, ion velocity, v, ion density, N and mean molecular weight, ¡, are shown along the inbound and outbound paths of the spacecraft

log R

Fig. 37. Results of a 3D-MHD model for the ion tail of comet Halley during Giotto encounter [210]. Ion, Ti, and electron temperatures, Te, magnetic field, B, ion velocity, v, ion density, N and mean molecular weight, ¡, are shown along the inbound and outbound paths of the spacecraft

Fig. 38. Schematic sketch of the folding of the solar wind magnetic field lines around a comet [5]

Distance lo Comet (km)

Fig. 39. Profile of the electron temperature used in various model calculations [97]

Distance lo Comet (km)

Fig. 39. Profile of the electron temperature used in various model calculations [97]

The size of the interaction region of comets with the solar wind can be approximated by the stand-off distance of the bow shock, RI. It depends on the solar wind flux, p©u©, the cometary gas production rate, Q, the average particle mass, mC, the neutral gas speed, u, and the ionization rate, k:

Ri, therefore, scales with the gas production rate for given solar wind conditions. We note, however, that in weakly active comets or at large heliocentric distances, no bow shock will form. Weak comets may also lack a diamagnetic cavity around the nucleus, and the solar wind magnetic field and solar wind particles may even penetrate to the nucleus surface.

Good knowledge of the temperatures is crucial to understand the ion chemistry in the inner coma (see Sect. 5). Unfortunately, the temperature of the electrons, Te, in the energy range relevant for electron recombination -which is an important loss process for ions in the inner coma - could not be measured in situ so far. Several attempts have been made to derive the temperature distribution from measurements of the ions in Halley's coma by various models (Fig. 39) with different results. They all agree on a steep decrease of electron temperature in the inner coma, because in the diamag-netic cavity region, electrons are cooled efficiently by collisions with water molecules. However, where the steep decrease in Te occurs is difficult to determine. [98] derived a distribution for Te (Fig. 39) that could match well the measurements of H3O+ ions in comet Halley (see Sect. 5). Spatial mapping of HCO+ ions in comet Hale-Bopp (e.g., [147, 148]) showed a reduced column density in the inner coma, which could also be explained by low electron temperatures leading to increased loss processes [177]. Except for such indirect evidence, the low electron temperature range is difficult to measure, and we have to wait for future space missions for in situ data.

Observations of Ion Tails

Observations of ion tails have shown that they always point almost radially away from the Sun, with only a slight abberation angle of a few degrees. On the basis of the appearance of cometary ion tails, Biermann concluded in 1951 [22] that a flow of charged particles must exist streaming radially away from the Sun, the solar wind. This may have been the most important implication of observations of cometary ion tails for our understanding of the solar system. In addition, cometary ion tails serve as a laboratory for plasma phenomena, which are difficult to simulate in a laboratory on Earth.

Our today's general picture of the comet-solar wind interaction has been confirmed by the ICE and Giotto spacecrafts visiting comets P/Giacobini-Zinner, P/Halley, and P/Grigg-Skjellerup (e.g., [12,152,165]). For example, magnetometer measurements showed the folding of the magnetic field lines around comet Halley and the existence of a diamagnetic cavity [165]. The results of the plasma experiments on Giotto are summarized in numerous reviews and books, for example [151], and it would require too much space to summarize even the most important measurements here.

On a large scale, ion tails show many highly time variable phenomena, such as rays folding toward the main tail, disconnections of the whole tail from the nucleus region, formation of clouds moving down the tail, etc. It has often been proposed that the response of the cometary ion tails to changes in the solar wind can be used as a tracer of the solar wind conditions. Observations of ion tails and model simulations to understand structure formation (e.g., [168, 179,193,211]) have provided considerable improvements in our understanding on the comet-solar wind interaction. However, it has also become evident, that a one-to-one correspondence of ion tail structures to solar wind features may be an oversimplifying assumption. Different conditions in the solar wind can lead to similar appearances of the ion tails, e.g., the formation of tail rays or disconnecting clouds, making the identification of the origin of ion tail variations in the solar wind difficult.

4 Emission Excitation in the Gas Coma

The molecules, atoms, and ions in the cometary coma are visible through their emitted radiation. Higher energy levels are excited by, for example, solar energy or collisional excitation. Transitions can occur among rotational levels within a vibronic band (pure rotational transitions), among rotational levels of different vibrational bands (ro-vibrational transitions) and ro-vibrational levels in different electronic bands (electronic transitions) (Fig. 40). Allowed transitions between energy levels with wave number v depend on the relevant transition levels:

v « (E'ei - El) + (E;ib - EVib) + (E'rot - EOt) (25)

Fig. 40. Schematics of energy levels of a hypothetical molecule. Two electronic energy levels are shown with two vibrational energy levels each. For the vibrational levels, several rotational energy levels, are indicated

Emissions of electronic transitions (E'ej — E^j) occur in the UV, optical, and the near-infrared range. Emissions in the UV are not transmitted through the Earth atmosphere and can be detected only by rockets and from spacecrafts. This is the case for most atoms and atomic ions in cometary comae. Vibrational transitions (E'Yib — EVib) are observed in the infrared wavelengths range. In the Earth atmosphere, water molecules are efficient absorbers of IR radiation, and therefore, detections of cometary emissions from the ground are possible only in atmospheric windows free of water absorption bands. However, infrared space telescopes allow us to observe emissions over a wide wavelengths range. Pure rotational transitions (E'Iot — E"ot) are observed at radio wavelengths. It is beyond the scope of this introduction text to explain in detail molecular excitation, and we refer to the standard literature [106-108].

The wavelengths range at which emissions are primarily observed differs for parent molecules and daughter radicals. The lifetime of electronic transitions, t <x. 1/Auj (Auj: Einstein A coefficient of the transition) is in the order of 10~8 s. This is much shorter than the lifetime of vibrational (t = 10~3 s) or rotational (t = 1s) transitions. Therefore, depending on the energy levels excited, the observed emission of a molecule is found in different wavelengths ranges:

• Atoms: They emit by electronic transitions at UV and optical wavelengths.

• Daughter radicals are observed mainly at optical wavelengths, because solar photons excite their upper electronic bands, which have very short lifetimes.

• Parent molecules: High energy photons in the optical to UV range lead to fast photo-dissociation of the molecule. Therefore, emission excitation occurs instead by collisional and radiative excitation of the lowest rotational and ro-vibrational energy levels. At radio wavelengths, for example, we observe rotational transitions in the lowest vibrational bands. • Symmetric parent molecules: Symmetric molecules without permanent dipole moment are a special case because for them pure rotational transitions are not allowed, and they can therefore only be observed by their ro-vibrational transitions at IR wavelengths, but not in the radio range (examples: CH4, C2H2, C2H6).

In the UV range atoms such as H, O, C, and S are detected (see overview by [75]), as well as parent molecules such as CO (the so-called Fourth positive system at 1450 Â and the CO Cameron bands at 2050 A indicating CO2 photodissociation) and the S2 molecule (see overview by [32]).

Observations of comets in the optical wavelength range have the longest history and statistical baseline of cometary observations. Therefore, this wavelengths range is important when statistical comparisons between comets are made, although observations at longer wavelengths ranges (IR, radio) increase in modern times with improved instruments. Furthermore, the high fluorescence efficiency of some molecular emissions, for example CN, make optical emissions an ideal tracer of gaseous activity in faint comets and comets at large heliocentric distances.

Transitions of several daughter radicals are well known in comets, as shown in Fig. 41. The transitions appear as band sequences in low-resolution spectra, because emission lines with the same Av between upper and lower vibrational

4000 5000 6000 7000 8000 9000

Wavelength [A]

Fig. 41. Optical spectrum of comet 9P/Tempel 1. The observations were made on July 3/4, 2005, at ESO. The comet was at rh = 1.6 AU and ^ = 0.9 AU, respectively. Gray: Spectrum with night sky subtracted. The underlying continuum caused by scattered solar light on cometary dust particles can be seen. Black: the same spectrum, but continuum subtracted

4000 5000 6000 7000 8000 9000

Wavelength [A]

Fig. 41. Optical spectrum of comet 9P/Tempel 1. The observations were made on July 3/4, 2005, at ESO. The comet was at rh = 1.6 AU and ^ = 0.9 AU, respectively. Gray: Spectrum with night sky subtracted. The underlying continuum caused by scattered solar light on cometary dust particles can be seen. Black: the same spectrum, but continuum subtracted band, v, are very close in wavelengths and can be resolved only with highresolution spectroscopy. The most prominent emission bands in the optical range arise from CN, CH, C3, C2, NH, and NH2 molecules.

Cometary parent molecules are observed mainly by their rotational and ro-vibrational transitions in the radio and IR-domain. The most important species to be observed include the main cometary ices: H2O, CO, and CO2, in addition to a large number of minor species, such as C2H2, C2H6, CH4.

Because of the complex nature of some emission bands because of overlapping emissions of different species, an identification is sometimes difficult. For example, around 3.4 |m a broad emission feature is seen in many comets, e.g., in comet Halley [45]. The main emission is usually attributed to the C-H stretching vibration mode. It is, however, still unclear which molecules contribute to this emission. Parts are believed to result from methanol and formaldehyde, but other organic molecules may contribute. Recently, the CH-feature has been observed also by the Deep Impact spacecraft in comet Tempel 1 [1] in the ejecta material after the impact. The relatively strong feature after impact was preliminarily interpreted as evaporating organic material. Clearly, identifying the contributions to this emission band is an interesting future task, but deserves more modeling efforts of the excitation conditions, including optical depths effects and non-LTE (LTE: local thermo-dynamical equilibrium) excitation conditions.

Altogether, a large number of observations of cometary emissions exist from the UV up to the radio range. These emissions are used to spectroscopi-cally identify the species present in the coma and provide us with an inventory of cometary constituents. [75] provides a list of the spectroscopically observed daughter species in comets up to now, together with the wavelength/frequency of their main emissions. [32] discuss the detected parent molecules.

At present, 24 parent molecules have been detected by spectroscopic emission features. The presence of two parent molecules, CS2 and N2, is, however, inferred only by their daughter products CS and N+. In addition, some species believed to be parent molecules may instead be daughter products of more complex organic species. Such complex parent molecules have been suggested, for example, for CO and H2CO (see Sect. 5).

Emissions of the refractory component of comets (Ca, Co, Cr, Cu, Fe, K, Mn, Ni) can be detected mainly in sun-grazing comets, with the exception of Na, which could be observed in several comets up to now (e.g., [51,109,166, 176]). Therefore, sodium is one of the rare species that allows us to study the non-refractory component of cometary nuclei.

Many more molecular species have been searched for, but remained undetected. In particular, complex parent molecules are difficult to detect spectro-scopically because the intensity of the individual rotational and vibrational lines of these molecules is very low. A list of the undetected parent molecules searched for at radio wavelengths can be found in [55]. Line catalogs of highresolution spectra at optical wavelengths (e.g., [42]) show the well-known daughter radicals, but also contain a large number of unidentified emission lines. Although most of them are probably part of the band systems of the already well-known species, the possibility of yet unidentified molecules in comets contributing to these spectra cannot be ruled out. Comets are studied over a wide range of wavelengths. However, not the whole wavelength range is fully exploited yet by observations, and our inventory of cometary parent molecules is clearly not yet complete.

Indications for complex organic species resulted from mass spectrometer data obtained during the Giotto fly-by on comet Halley. The neutral and ion mass spectrometers on board Giotto gave the first in situ measurements of the volatiles in a cometary coma. Unfortunately, a clear identification of these high mass ranges is difficult. Polyoxymethylene (POM) molecules have been proposed to explain the regular mass peaks observed [111]. An overview of the results of the ion mass spectrometer can be found, for example, in [14].

4.1 Resonance Fluorescence

The dominant excitation mechanism for the electronic transitions observed in the optical, near-UV, and near-IR range is resonance fluorescence by the solar flux. The strength of an observed emission is calculated using the fluorescence efficiencies or g-factors (in units of [photons-1]). The g-factors are calculated from the absorption oscillator-strength, f, the Einstein A coefficients, Aik, and the solar flux, F\ at the observed wavelength, A [40]:

The g-factor calculated for the solar flux at rh = 1 AU scales with distance as r-2.

If an observed emission band is caused by pure resonance fluorescence and the relevant g-factors are known, the conversion from observed fluxes to molecular column densities, N, is straightforward:

4n 1

Here, F denotes the observed emission flux in the aperture and Q the aperture size used. Converting from column densities to molecular gas production rates of the comet then usually requires a model of the spatial distribution of the molecules in the coma (see Sect. 6).

The solar spectrum as seen by a coma molecule is Doppler-shifted because of the velocity of the molecule. The fluorescent excitation of emission bands near solar Fraunhofer absorption lines, therefore, can be a strong function of the heliocentric velocity component of a molecule. The Doppler shift has two components: the heliocentric velocity component of the comet's orbital velocity (Swings effect [201]), and the motion of the molecule with respect to the nucleus (Greenstein effect [92]). The latter is important only for fast moving atoms, such as Na, or ions, moving with velocities of several tenth to hundred km/s. Bands most strongly affected by the Swings and Greenstein effects are lying near strong solar Fraunhofer absorption lines. An emission band significantly affected by the Swings effect is, for example, the CN (0-0) band at 389 nm.

In addition, variations of the incoming solar radiation affect the excitation of emission bands. The solar flux arriving at a cometary molecule depends on the 11-year solar cycle. It may also vary with solar rotation and the occurrence of sunspots.

Fluorescence excitation models are usually made for observations near 1 AU, and their application to large heliocentric distances needs to be verified. The relative importance of collisional and radiative excitation processes in the coma can vary with heliocentric distance and cometary activity and can lead to deviations of the excitation observed from model predictions. For example, the relative band strengths of NH2 detected in comet Hale-Bopp beyond rh = 3 AU did not agree with the fluorescence excitation models known at that time [177]. New g-factors [128-130] gave agreement with the observations and should now be used.

4.2 Prompt Emission

Prompt emission means a daughter molecule or atom is formed in an excited state and then performs a radiative transition into the ground state. A well-known example for prompt emission in the optical range is the formation of oxygen by:

In this reaction, the oxygen atom is formed in an excited state (indicated by *). Observed emissions of oxygen are O 1D: 6300 A, 6364 A, and 1S: 5577 A. In case of prompt emission, every transition occurs only once, immediately after formation of the oxygen atom. This makes prompt emission an ideal tracer of the parent molecule H2O. Unfortunately, oxygen atoms are produced by the same process also in the Earth atmosphere. Therefore, high spectral resolution and a sufficiently large Doppler shift of the comet geocentric velocity component are needed to separate the cometary emission from Earth atmosphere contamination. Further complications arise, for example, when collisional quenching of the upper levels plays a role in very active comets.

Other examples for prompt emission are the CO "Cameron bands" at 2050 A, which are excited after formation by photodissociation of CO2, and H Lya emission. Images and spectra of H Lya have shown the enormous extent of the cometary neutral hydrogen coma (Fig. 42). Recent modeling of the excitation and dynamics of cometary hydrogen can be found in [46, 47] and [182].

Fig. 42. Image of the hydrogen coma of comet Hale-Bopp observed by Lya emission [48]. The white dot to the right gives the solar disc to scale. Inserted is a picture of the comet taken at optical wavelengths for comparison with the elongation of the ion and dust tails

4.3 Optical Depth Effects

Obviously, sunlight reaches the nucleus surface and sublimates the volatile ices. The cometary coma, therefore, is generally optically thin. However, optically thin conditions may not be fulfilled for strong resonance lines, such as H Lya. High optical depths in the coma will affect the resonance excitation of radiation in the coma as well as photochemical processes.

The optical depth is a function of wavelengths and position in the coma, R:

Here, Ni is the number density of molecules in the line-of-sight to the Sun, ji is the absorption cross section at wavelengths A.

An additional note is added here on optical thickness effects at visual and infrared wavelengths by light scattering on coma dust particles. This effect may be important for active dusty comets, as discussed in [161] and references therein. These models compute the ambient solar flux on the nucleus as input for surface sublimation models. We remark, however, that such effects can also influence the excitation of molecules very near the surface.

4.4 Excitation of Rotational and Vibrational Transitions

To calculate the population of the rotational levels, fluorescence equilibrium is often assumed throughout the outer coma. However, this assumption is not

fulfilled for short-lived molecules and in the inner coma where collisional processes are important. Depending on the molecules observed, both excitation mechanisms must therefore be considered. An overview of the excitation of rotational and vibrational transitions can be found in for example [56, 59] and [58].

Calculating the excitation by collisions requires knowledge of the colli-sional cross-sections of the molecules and the coma temperature. The thermal excitation of a molecular energy level is determined by the Boltzmann distribution:

No go

Here, Nu and N0 are the number of molecules in the excited and ground level, respectively. E is the excitation energy of the upper level. gu 0 are the statistical weights of the upper and lower levels. In general, the temperature profile in the coma is a function of nucleocentric distance, r, and depends on the radiative cooling and photolytic heating processes (see Sect. 3). The population of the ro-vibrational levels by collisions is therefore also a function of nucleocentric distance.

To determine the excitation of the molecules, we assume optically thin conditions and neglect the influence of dust particles. We can also assume that collisions are important only among neutrals because ion densities in the innermost coma are low. Furthermore, we assume radiative excitation occurs only between vibrational bands. The rate of collisions is given by:

Here, u is the mean velocity between two collisions. The relative excitation of a rotational level i in case of a non-steady state calculation is then given by:

Here, pij is a transition rate from level i to j, and pij = Cij + gij for Ei < Ej or pij = Cij + Aij for Ei > Ej [32].

In addition to this set of stiff differential equations describing the population of the rotational and vibrational levels, one needs a model of the spatial distribution of the molecules (see Sect. 3).

To derive the rotational temperature, Trot, from the population of excited rotational levels, several emission lines of a molecule (e.g., Fig. 43) need to be observed simultaneously (e.g., [30,164,209]). If the observed transitions are relaxing only slowly to fluorescence equilibrium, their population is determined by the collisions occurring in the inner coma, and Trot is close to the kinetic temperature, Tkin.

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