P Canizares

M Chmeissani

A Conchillo

M Diaz-Aguiló

E García-Berro

L Gesa

F Gibert

C Grimani

I Lloro

A Lobo
email: lobo@ieec.fcr.es

I Mateos

M Nofrarias

J Ramos-Castro

J Sanjuán

Carlos F Sopuerta

H M Araújo

P Wass

The LISA PathFinder DMU and Radiation Monitor

Keywords: LISA, LISA Pathfinder, gravitational wave detector, charge deposition, Radiation Monitor, Data Management Unit PACS numbers: 04.80.Nn, 95.55.Ym, 04.30

The LISA PathFinder DMU (Data Management Unit) flight model was formally accepted by ESA and ASD on 11 February 2010, after all hardware and software tests had been successfully completed. The diagnostics items are scheduled to be delivered by the end of 2010. In this paper we review the requirements and performance of this instrumentation, specially focusing on the Radiation Monitor and the DMU, as well as the status of their programmed use during mission operations, on which work is ongoing at the time of writing.

Introduction

LISA is a technologically sophisticated mission. So complex, in fact, that ESA decided to fly a precursor mission to ensure technology readiness and maturity have reached a safe status to start LISA. Launch for PathFinder is currently set for early 2013.

Confidential: not for distribution. Submitted to IOP Publishing for peer review 10 January 2011 LISA PathFinder (LPF) has a reduced acceleration noise budget, both in magnitude and in frequency band with respect to LISA [START_REF] Vitale | Science Requirements and Top-level Architecture Definition for the LISA Technology Package (LTP)[END_REF]:

S 1/2 δa,LPF (ω) ≤ 3×10 -14 1 + ω/2π 3 mHz 2 ms -2 / √ Hz (1)
in the frequency band between 1 mHz and 30 mHz. This noise can be apportioned to different components, each of them individually interfering with the mission performance. Requirements are set on each of those so that the total sum is compatible with equation (1) [START_REF] Vitale | Science Requirements and Top-level Architecture Definition for the LISA Technology Package (LTP)[END_REF]. Thermal, magnetic and charged particle flux are specifically monitored by dedicated hardware and software, forming what is known as the Data and Diagnostics Subsystem (DDS), which also includes the DMU (Data Management Unit), the LTP (LISA Technology Package, or mission payload) computer. The DDS has been designed and built in IEEC-CSIC in Barcelona, Spain, and Flight Models of each of its parts have been delivered, or will be by the end of 2010, after passing extensive tests to ensure their performance during mission operations.

In this paper we review the latest progress with the DDS parts, specially focusing on the Radiation Monitor and the DMU, which have been the subject of our work during the last few months. We will however give a quick summary of the other diagnostics items as well in the first sections; a more detailed description of them can be found in reference [START_REF] Cañizares | [END_REF].

Thermal diagnostics

The thermal stability in the LTP Core Assembly (LCA) has to meet a very stringent requirement: A set of 24 temperature sensors are scattered around the LCA, figure 1, which are intended to measure temperatures at as many strategic spots. More specifically, in the outer walls of the Electrode Housings of both Test Masses (TM), at both Optical Windows (OW), at the Optical Bench (OB) and in the suspension struts (SS). In order to be able to make significant measurements in an environment with such a degree of thermal stability as required by equation ( 2), an even more demanding requirement must be set on the performance of these sensors. This is

S 1/2 δT (ω) ≤ 10 -4 K Hz -1/2 , 1 mHz ≤ ω 2π ≤ 30 mHz (2)
[3] S 1/2 δT , sensors (ω) ≤ 10 -5 K Hz -1/2 , 1 mHz ≤ ω 2π ≤ 30 mHz (3)
The temperature sensing in the DDS is done by means of NTC (Negative Temperature Coefficient) thermistors, because they have more pronounced slopes in the temperature-resistance plane than other devices such as Platinum resistors. In addition, an extremely quiet front-end electronics must be used to drive them and acquire data. This electronics is part of the DMU, and Figure 2 shows the performance of a differential temperature measurement between two NTCs as recorded in the latest tests done with Flight Model sensors and DMU. This was April-2010. A detail must however be clarified: because of mass and power budget limitations, not every one of the 24 thermistors can have its own electronic board; rather, they are multiplexed in six groups of four. While this makes the system globally compliant with those constraints, it introduces additional noise, as the real sampling frequency is divided by the multiplexing factor in each thermistor. This is however not 4, as one would infer, but 6 because an additional differential temperature plus a control measurement are also multiplexed in each block. Therefore the multiplexed system noise is higher by a factor of 6 1/2 2.4 than shown in Figure 2. Note however that we are still fully within requirements. Actually, the (individual) NTC noise is below 10 -5 K Hz -1/2 all the way down to 10 -5 Hz, i.e., well within the LISA band. This means the current LTP thermal system can probably be transferred basically as is into LISA.

Precision heaters

It needs to be stressed that such marvellously working temperature sensors are not by themselves of much use for the mission. As explained in the previous section, we need these monitors to record temperature fluctuations which affect the performance of the LTP. But how do we link a (known) temperature fluctuation with an (unknown) TM shake or an interferometer phase jitter? Although this link can be modelled, it has to also be established in flight by direct measurement -since the model may be inaccurate.

To this effect, a set of 14 precision heaters are also installed in the LTP at some suitable spots [START_REF] Lobo | LTP thermal response to periodic signals injected by heaters[END_REF]. Their role is to inject controlled thermal signals in the system such that they are strong enough to be clearly seen in the LTP readout, i.e., with a signal-to-noise ratio around 100. Temperature measurements are taken as well and hence a relationship is established between the latter and the observed response of the interferometer and/or the Gravitational Reference Sensor (GRS). The analysis is done in frequency domain, so that a set of transfer functions is obtained relating temperature variations to LTP response. Finally, it is assumed that the behaviour of the LTP is maintained when the heaters are switched off.

Heaters are driven with very low noise DMU electronics. There are two major groups of heaters: those in the GRS and those elsewhere [START_REF] Lobo | LTP thermal response to periodic signals injected by heaters[END_REF]. The ones in the second group are rather conventional kapton heaters, consisting in a wire winding held together by an elastic resin. These heaters, however, cannot be attached to the walls of the Electrode Housing (EH) of the GRS due to their magnetic properties: they actually have some significant quantities of ferromagnetic elements, notably Nickel, which are likely to compromise the stability of the TM. It was therefore decided that NTCs should also be used as heaters in that part of the LTP.

These NTCs are a bit more difficult to drive than the others due to their changing electric resistance upon being heated up. Indeed, when a voltage is applied to an NTC the current through it will heat it up, hence its resistance will drop, hence the current and power it dissipates will increase. It has been shown [START_REF] Sanjuán | Development and validation of the thermal diagnostics instrumentation in LISA Pathfinder[END_REF] that, under suitable conditions of initial NTC temperature, thermal resistance between the heater and the attachment block, and voltage range, this process ends up in a stationary state in a matter of about 10 seconds. This is good enough for the time scales of the heating operations (normally a few thousand seconds), but the DMU has to be programmed so that these heaters be commanded with a voltage V such that, in the stationary state, the power dissipated in the heater, P ∞ , is precisely the required one. The relationship between the voltage applied and the stationary state power dissipation turns out to be somewhat involved:

P ∞ = V 2 R(T ∞ ) = V 2 R T 0 + θV 2 R 0 + α 0 θV 2 (4)
where T 0 is the NTC temperature just prior to activation (measured with an adjacent sensor), θ is the above mentioned thermal resistance (K/W), and α(T ) = R -1 dR/dT is the thermistor's temperature coefficient. The dependence of R on T is usually given by the Steinhart-Hart equation [START_REF] Steinhart | Calibration curves for thermistors[END_REF], again a somewhat involved mathematical expression. [In equation ( 4

), R 0 ≡ R(T 0 ) and α 0 ≡ α(T 0 )].
The algebraic complexities of the above formulae make it impossible to invert equation ( 4) to obtain the voltage V as a function of the power P ∞ in a closed-form expression. This circumstance has motivated the use of Look-Up Tables (LUT) in the DMU software which commands the GRS heaters: for a given thermal resistance θ (measured on ground) and an NTC temperature (measured dynamically on flight), the SW looks for the tabulated power which is closest to the one required for injection, then reads off the LUT the activation voltage, and applies it [START_REF] Conchillo | LTP Electrode Housing Heaters commanding[END_REF].

Magnetic diagnostics

The LTP Test Masses are made of a 73 % Au + 27 % Pt alloy, a good combination to keep as low as possible their magnetic properties and, at the same time, to provide sufficient mechanical robustness to withstand launch shaking while caged. It is however impossible to avoid the presence of ferromagnetic residuals after the casting, hence some magnetic remanence will be there. Limits have been set on remnant magnetic moment m 0 and susceptibility χ as [START_REF] Cañizares | [END_REF] |χ| < 2.5 × 10 -5 , |m 0 | < 2 × 10 -8 Am 2

(5)

These parameters couple to the surrounding magnetic field, mostly created by spacecraft electronic boxes and other components, such as solar panels, FEEP (Field Effect Electric Propulsion, the satellite's micro-thrusters), etc., thereby creating forces and torques on the Test Masses, and fluctuations thereof, i.e., magnetic noise. According to standard electromagnetic theory, forces and torques are respectively given by

F = M + χ µ 0 B •∇ B V (6)
and

N = M × B + r × (M•∇) B + χ µ 0 (B•∇) B V (7)
The meaning of symbols in the above formulae is as follows: Vacuum magnetic constant (4π×10 -7 m kg s -2 A -2 ) while • • • indicates TM volume average of enclosed quantity. For example, for any magnitude f (x), such average is defined by the volume integral

B
f ≡ 1 V V f (x) d 3 x (8)
over the TM volume. Magnetic noise can be readily inferred from the above formulae.

There is no specific requirement on either magnetic field or gradient fluctuations, there is only one on the total contribution of magnetic disturbances to the overall mission acceleration noise budget; magnetic is required to stay below 40% of that budget [START_REF] Vitale | Science Requirements and Top-level Architecture Definition for the LISA Technology Package (LTP)[END_REF].

On the other hand, there are specific requirements on the DC values of magnetic field and gradient, which should be kept below 10 µ-Tesla and 5 µ-Tesla/m, respectively. This is because of the quadratic dependence of the force F on the magnetic field B, due to the presence of a non-zero susceptibility -see equation ( 6); when it comes to evaluating the fluctuations, it is immediately seen that DC field values couple to gradient fluctuations, and also DC gradient values couple to field fluctuations. Magnetic fields in the LTP are monitored by means of four tri-axial fluxgate magnetometers, located in the periphery of the LCA. These are not ideally suited to infer the magnetic field and gradient at the TMs, due to their distance to the latter. Classical interpolation methods have failed to produce acceptable results, but neural network algorithms have been developed and studied at IEEC which produce significant improvements in our ability to determine field values at the TMs [START_REF] Diaz-Aguiló | [END_REF]. The reader is also recommended to look into Marc Diaz Aguiló's contribution to this Proceedings for more in-depth analysis and details.

Alternative solutions

While it is not possible to make changes now to the general LPF plan, e.g., magnetometers cannot be replaced, it would be unwise to think of LISA carrying on board a set of a few high performance but distant and voluminous fluxgate magnetometers. Ongoing research at IEEC has revealed that alternatives to the LPF scheme seem to exist. These are based on both a new type of sensors and driving electronics. The new sensors are tiny AMR (anisotropic magneto-resistor), while the electronics resorts to periodic flipping voltages to apply set-reset signals to the devices, thereby enhancing their performance. In Figure 4 we see noise curves of various magnetic sensors and activation techniques which show the significant advantage which can be potentially drawn from the new AMRs. There are issues still to be properly addressed, perhaps most notably the possible magnetic back-action of the sensors, since they have tiny ferromagnetic cores. Analysis so far point towards a negligible such back-action, the reader will find much more detailed information in Ignacio Mateos's contribution to this Conference Proceedings. Tesla Hz -1/2 is the current LPF magnetometer noise requirement, which is comfortably met in all cases considered, i.e., for frequencies above 1 mHz. The curves show that the best performance corresponds to the AMR with a flip voltage scheme, showing significant improvements over the more conservative lock-in approach. The Fluxgate performs well but is slightly noisier at LISA frequencies.

Control coils

Just like thermal diagnostics require precision heaters, so magnetic diagnostics require precision induction coils. Their purpose is very similar to that of the heaters, though with a difference: strong magnetic signals injected by the induction coils, together with the LTP response to them (again in the order of SNR 100) can be used to determine the remnant magnetic moment and the susceptibility of the TMs. More information can be found in [START_REF] Cañizares | [END_REF].

The Radiation Monitor

LPF will be stationed for operations in a Lissajous orbit around the Earth-Sun Lagrange point L1. This is about 1.5 million kilometres from Earth, i.e., well beyond its radiation belts. The spacecraft will thus be exposed to particles in the solar wind, but also to others, mostly of galactic origin. The composition of these fluxes in charged particles is roughly 90% protons, 8% Helium ions, and 2% heavier nuclei (Carbon and higher Z) and electrons. These will charge the spacecraft through both direct and indirect deposition, since secondary particles are generated as the primary ones travel across the various materials. What creates concern about charging is actually only the TMs, as charge deposition in them generates spurious potentials, hence noise in the GRS readout due to the random character of those depositions. Extensive simulations done at Imperial College [9] with the public CERN tool Geant-4 show that only primary particles with energies roughly above 70 MeV can possibly make it to the TM; lower energy ones are deterred by the satellite structure surrounding the TMs, and hence can cause no harm to the experiment. The simulations also show that the charging rate of the TMs depends on whether the primary particles are Galactic Cosmic Rays (GCR) or correspond to flares in the Sun (SEP events, Solar Energetic Particles ‡). These two types of flux have different energy spectra, and the way to distinguish between them is therefore to do spectroscopy. Figure 5 shows quite clearly the above distinction: spectra of GCR are displayed on the left and deposition rates in the right panel. Data are due to Araújo et al [10]. See also Catia Grimani's contribution to this volume.

The Radiation Monitor (RM) role in LPF is to identify the charging rates in the TMs by short term monitoring of the charged incoming particles. For this, a sensor consisting in two PIN diodes in telescopic configuration is used. The PINs are enclosed in a copper shielding which prevents primary particles with energies below 70 MeV from reaching them, thereby recreating the actual situation in the TMs, see Figure 6, left panel. An ionising particle passing through the reverse-biased PIN junction will release many charge carriers (1 electron/hole pair per 3.6 eV deposited) which is collected by a charge amplifier. The signal is then processed by suitable a electronics, which is able to: 1) identify single particle arrivals in the PIN, well above 6 000 per second, the largest SEP shower ever observed, and 2) determine the energy deposition spectrum of an incoming particle flux [START_REF] Araújo | Note on the design of a particle monitor for LISA Pathfinder[END_REF]. Because the PINs are sensitive to photons, too, but these may not deposit charge in the TMs, the RM makes use of two PINs such that spectral information is only delivered when events are detected in coincidence in both PINs, thereby filtering out photon events, which would only be seen in one, where it would be absorbed, hence never reaching the other. RM data are stored in a histogram format as shown in Figure 6, right panel.

One such histogram is the RM telemetry unit. Bins are scanned and filled at 100 Hz. Data are accumulated in the DMU during 600 seconds (i.e., 10 minutes) and then sent to the On Board Computer (OBC) to be telemetered to Earth according to mission schedules. Data are subsequently analysed off-line to look for correlations with the Charge Management System, which takes much longer periods to produce data. We can thus use the RM to detect shorter term charging fluctuations as well as modulations which should be useful to more thoroughly assess the whole process [START_REF] Shaul | Analysis of Data from the Radiation Monitor on LISA Pathfinder[END_REF].

At the time of writing, the LPF RM Flight Model is a few weeks from final delivery. Many tests have been done on it before that: extensive electronics tests, magnetic moment measurements, vibration, temperature stability of the RM performance and proton irradiation. The latter was carried out at the Paul Scherrer Institute (Switzerland) where a proton beam facility was used to generate low dose fluxes and assess the RM response for various energies and beam incidence angles. The prototype was also submitted to such test, except it was much more aggressive at that time, as many more things had to be checked then. Now we are reassured everything is in place and the test can be milder to avoid deterioration of the device before flying. Analysis of the irradiation results will soon be complete.

The DMU

The Data Management Unit (DMU) is the LTP computer, and has been designed, developed, manufactured and delivered by IEEC, Barcelona, on 11 February 2010. The DMU has three electronic boards: the Power Distribution Unit (PDU), the Data Acquisition Unit (DAU) and the Data Processing Unit (DPU), each of them duplicated (redundant) to cover potential failure in flight. Redundancy is however not full in the DAUs, where some diagnostics items are connected to only one DAU, see Figure 7, again due to mass-energy quotas. Protection against possible DAU failure has been maximised by careful distribution of each DAU's tasks.

The DMU fully controls all the diagnostics items functionality, including activation, acquisition, and telemetry as described in sections 2 and 3 above. With the exception, however, of the Radiation Monitor, which internally acquires its data and communicates with the DMU via a serial RS 422 line. But the DMU has many more interfaces with other parts of the LTP, as shown in Figure 7. Communications with the OBC and non-diagnostic parts of the LTP is done via two MIL-STD-1553 however, have also been passed by the rest of the DDS hardware, they are indeed not specific to the DMU.

The DMU Software

The DMU requires software to implement its functionality and to communicate with its various interfaces. This software has been created from scratch at IEEC, too, and has two main bodies: the Boot Software (BSW) and the Application Software (ASW).

The BSW is a minimal, non-real time operating system, which occupies 61 kilobytes of memory. It contains some 30 000 lines of C code, plus about 1 000 of Assembler code, and it was eventually burnt to a PROM (Programmable Read-Only Memory) chip, of course after extremely rigorous tests were passed, as the PROM cannot be either rewritten or replaced. The BSW starts the DMU and acts as scheduler, interrupt control and error detection and correction. A most important task for the mission which is also in charge of the BSW is to load the ASW, which understands and executes all the telecommands which are received from the Mission Operation Centre (MOC) following the mission Master Plan. The BSW can also patch the ASW should there be an upgrade deemed necessary for the continuation of the mission operations. Figure 9, left, graphically summarises these tasks. The BSW has long been burnt to an FM PROM, hence is fully ready for flight.

The ASW is built on top of RTEMS (Real-Time Executive for Multiprocessor Systems), a real-time operating system, in order to cope with the hard real-time constraints (i.e., task deadlines must be met on time) of the mission. The ASW, as already mentioned, manages all the scientific and technological functions of the DMU. It supports real time tasks up to a maximum frequency of 100 Hz, and is encoded in some 70 000 lines of C code, plus 300 of Assembler. In addition, it sends telemetry to the OBC via a MIL-STD-1553 bus. The ASW is stored in EEPROM (Electrically Erasable Programmable Read-Only Memory), which makes possible to receive upgrades from ground if necessary. It is executed from RAM. Stress tests done on the ASW show that it produces a maximum CPU load of 86%, therefore a bit tight on margin. Work is in progress to bring down that ratio by a few percentile points.

The ASW tasks are graphically summarised in Figure 9, right. In addition to diagnostics control, as already mentioned, it performs several operations and interfaces with other LTP subsystems. For example, it receives phasemeter data which analyses by single bin Fourier transform to produce feedback to the laser control system (e.g. frequency stabilisation). It also interacts with the caging mechanism, charge management device and Gravitational Reference Sensors front end electronics.

In addition to the mentioned stress tests, the ASW has undergone unit testing and validation, the latter by an independent entity which produced test scripts to check nominal test cases. Errors were filed and reported for debugging. The latest validated version is 2.3. ASW 2.4 was released in mid August 2010, with all functionality in it. By the end of 2010 a validated and optimised version of ASW 2.4 will be ready.

The SW is a dynamic component of the DMU, and work on it is therefore envisaged all the way to mission completion.

Conclusions

We have briefly reviewed here the essential parts, as well as their functions, of the DDS. As shown, the Spanish hardware contribution to LPF is really complete, up to minor changes/details. The IEEC team is still active, and will be until the end of the mission. New tasks, not reported here, include the preparation of the mission operations, software (not to be confused with the DMU software) modelling of the system response to validate the PORs (Payload Operations Requests) which constitute the basis for the entire mission Experiment Master Plan, and which are being prepared in the Science and Technology Operations Centre (STOC) in ESAC, near Madrid. In addition, on-line response and off-line data analysis of the mission yield has a lot to do with the diagnostics subsystem, hence IEEC is very actively involved in these matters, too.

Figure 1 .

 1 Figure 1. Schematics of the distribution of thermal diagnostics items, i.e., temperature sensors and precision heaters. Nomenclature: triangles represent heaters and are labelled H1-H14; squares represent temperature sensors and are labelled TS1-TS24; GRS-1 and GRS-2 are the gravitational reference sensors, OW-1 and OW-2 the corresponding Optical Windows, and OB is the Optical Bench. Heaters H9 through H14 and sensors TS17 through TS22 are attached to the LTP suspension struts -see more details of these structures in figure 3.

Figure 2 .

 2 Figure 2. Flight model temperature sensor behaviour. The plot corresponds to a differential measurement between two NTCs. As can be seen, it is fully compliant with the requirement in equation (3).

Figure 3 .

 3 Figure 3. LTP diagnostic magnetometers and induction coils. For clarity, the former are shown levitating, but they are actually bolted to the surrounding cylindrical wall with the shown braces.

  Magnetic induction field in the TM M Density of magnetic moment (magnetisation) of the TM r Vector distance to the TM centre of mass V Volume of the TM χ Magnetic susceptibility of the TM µ 0

Figure 4 .

 4 Figure 4. Magnetometers sensitivity curves. The horizontal, thicker grey line at 10 -8 Tesla Hz -1/2 is the current LPF magnetometer noise requirement, which is comfortably met in all cases considered, i.e., for frequencies above 1 mHz. The curves show that the best performance corresponds to the AMR with a flip voltage scheme, showing significant improvements over the more conservative lock-in approach. The Fluxgate performs well but is slightly noisier at LISA frequencies.

Figure 5 .

 5 Figure 5. Left panel: spectra of primary energies in a giant SEP event (September 1989) and of a normal one (May 2001) as obtained from the GOES Observatory data, and fitted. Also shown is a GCR proton spectrum taken at solar maximum. Right panel: charging rates in the TMs for a "normal" SEP event and GCR, also at solar maximum.

Figure 6 .

 6 Figure 6. Left panel: schematics of the Radiation Monitor. Right panel: the Radiation Monitor data histogram, including 1024 energy bins for coincident events and 40 bins for singles counts -see text for more details.

Figure 7 .

 7 Figure 7. Schematics of the DMU boards and their LTP interfaces.

Figure 8 .

 8 Figure 8. The DMU in a test facility at ESTEC. It is the dark box at the centre of a Helmholtz coil set, which shields the DMU from external disturbances during magnetic moment measurements.

Figure 9 .

 9 Figure 9. Conceptual diagram of the DMU software: BSW on the left and ASW on the right.

‡ The solar wind contains mostly keV particles, only a relatively small fraction are above 100 MeV, hence the name SEP.

Acknowledgments

Financial support from the Spanish Ministry of Innovation and Research (MICINN), contract ESP2007-61712, is gratefully acknowledged, as well as AYA08-04211-C02-01. CFS acknowledges support from the Ramón y Cajal Programme of the same Ministry, and a Marie Curie International Reintegration Grant (MIRGCT-2007-205005/PHY) within the 7th European Community Framework Programme, and PC acknowledges an FPU PhD Grant from MICINN. Part of this work was also supported by AGAUR, Generalitat de Catalunya.


C Grimani
email: catia.grimani@uniurb.it

H M Araújo

M Fabi

A Lobo

I Mateos

D N A Shaul

T J Sumner

P Wass

Galactic cosmic-ray energy spectra and expected solar events at the time of future space missions

Keywords: Cosmic rays, Sun: solar-terrestrial relations, Space interferometers

Galactic cosmic-rays (GCRs) and solar energetic particles (SEPs) affect observations on board long-lived space missions. We developed a parameterization of proton and helium fluxes for various levels of solar modulation during opposite polarity periods. In addition to long-term variations (decades), short-term fluctuations (minutes to days) were considered as well. In particular, we focused on data from experiments carrying magnetic spectrometers in space. The shortest GCR variations we were able to study are of the order of hours. We point out that GCR variations and fluctuations are strongly energy dependent. Energy dependent is also the detector charging on board space experiments. The measurements of energy differential fluxes and their variations are needed in order to evaluate properly the performance of future space missions. We present here the projections for the GCR fluxes and solar events at the time of LISA (Laser Interferometer Space Antenna) Pathfinder (LISA-PF).

Introduction

Solar activity level, drift of opposite charge particles in the global solar magnetic field (GSMF) and interplanetary processes affect cosmic-ray observations in the local interplanetary medium. Experiments devoted to cosmic-ray physics aim to infer from near-Earth measurements the interstellar spectra. In all other missions energetic particles might affect the performance of the onboard detectors. In LISA-PF and LISA, for example, solar and galactic proton and helium particles above 100 MeV per nucleon (MeV /n) limit the mission performance charging the onboard test masses (see for details [1]- [2]). In particular, short-term GCR fluctuations generate spurious signals in the experiment band [START_REF] Shaul | AIP Conf. Proc[END_REF] and SEPs associated with strong solar events overcome the whole mission noise budget in the low frequency range (see for example [START_REF] Grimani | [END_REF]).

Predictions of future solar cycle amplitudes allow us to estimate the intensity of GCR and the number of expected solar events during the next decades. The sunspot number is the most widely used proxy for solar activity prediction. In this work we adopt the projections of the next solar cycle intensity based on the observed trend of the solar spot number during the first months of this year [5]- [6]. According to these predictions, we estimate the GCR energy spectra and the number of solar events at the time of LISA Pathfinder at the end of 2012. The number of solar events were estimated according to the Nymmik's model [START_REF] Nymmik | Proc. 26th Int. Cosmic Ray Conf[END_REF]- [START_REF] Nymmik | Proc. 26th Int. Cosmic Ray Conf[END_REF]. Details of the LISA-PF mission are reported in [START_REF] Vitale | [END_REF].

Part of this work was carried out earlier, however an update was needed since the actual trend shown by the initial rise of the solar cycle 24 appears weaker than projections available up to 2008 [10].

Various theoretical models were proposed in the literature to take into account the effects of solar modulation and solar polarity on GCRs (see for example [11]- [12]). At this time we prefer an empirical approach based on data gathered by experiments carrying magnetic spectrometers in space that must be used to calibrate theoretical models. Proton, helium and electron data were considered. The method was discussed accurately in [START_REF] Grimani | Proc. 30th Int. Cosmic Ray Conf[END_REF]. Our predictions will be redundantly tested by cosmic-ray experiments in flight at the time of LISA-Pathfinder such as AMS [START_REF] Ting | Europ. Cosmic Ray Sympos[END_REF] and by the onboard monitors of incident solar and galactic proton and helium nuclei above a few tens of MeV/n [START_REF] Mateos | [END_REF].

GCR short-term fluctuations are discussed here in addition to long-term variations. Among short-term fluctuations we include the 27-day variations related to the Sun rotation, Forbush decreases (see Section 3.2 below) and variations of the order of hours. Cosmic-ray fluctuations could not be studied in smaller intervals of time. This limitation arises from the large statistical uncertainties affecting differential flux measurements carried out by experiments with small geometrical factors resulting from the use of magnetic spectrometers.

For completeness we add that studies of GCR fluctuations down to minutes were carried out, for example, by Starodubtsev, Usoskin and Mursala [16] using ground neutron monitors.

Solar cycle 24 projections and observed initial rise

Predictions of a solar cycle include both amplitude and timing. Timing depends on the characteristics of the solar minimum. Due to the unusual long duration of the last solar minimum we presently expect the next maximum to occur in 2013. Present projections for the solar cycle 24 are reported in the right panel of fig. 1 [5]- [6]. Solar cycle 24 projections available up to 2008 according to Hathaway and Dikpati [10] appear in the left panel of fig.

1. An extensive review of the solar cycle 24 projections is reported in [17]. Unfortunately, the majority of these predictions were very different from the actual trend shown by the first phase of the solar cycle 24. In particular, out of 54 predictions, 4 only reported a solar maximum to occur beyond 2012. Intensities ranging between 185 and less than 40 for the average annual sunspot number (R 24 ) were proposed. However, 78% of the whole sample of predictions indicated R 24 above 100 at the maximum. In other words, almost the totality of predictions presented a solar cycle of mediumstrong amplitude. Conversely, the anomalous long duration of the last solar minimum and the trend of the rise of the solar cycle 24 indicate approximately R 24 =77±20. We point out that a quite good projection was reported by Kontor [START_REF] Kontor | Statistics based regular extrapolation[END_REF] for both R 24 (70±17.5) and timing (2012.96). Moreover, Li, Gao and Su [START_REF] Li | [END_REF] indicated an average annual sunspot number of 80 and the maximum to be reached in February 2013 (±8 months) in case of a slow riser cycle.

Galactic cosmic-ray proton and helium energy spectra at the time of LISA-PF

Assuming that LISA-PF will be launched during the first quarter of 2012 and that about 3 months will be needed to reach its final orbit in L1, data will be taken during the second half of 2012, near the next solar maximum and possibly during the same negative polarity epoch we are presently experiencing. We recall that the GSMF polarity is positive (negative) when solar magnetic field lines are directed outward (inward) from the Sun northern pole. Galactic cosmic-ray observations near Earth are affected by both solar activity level and solar polarity [20]. The effect of the solar modulation on GCRs during positive polarity periods is well represented by the symmetric model in the f orce f ield approximation by Gleeson and Axford [21]. This model (see eq.( 1)) allows us to estimate through an energy loss parameter, Φ, the fluxes of cosmic rays at a distance r from the Sun, at the time t (J(r, E, t)) assuming time-independent interstellar fluxes (J(∞, E + Φ)). In eq.( 1) E Table 1: Proton (p) and helium (He) flux parameterization at the interstellar medium.

A P 1 P 2 p (1.94 ± 0.13) × 10 4 0.70 ± 0.52 2.76 ± 0.03 He (7.10 ± 0.56) × 10 3 0.50 ± 0.31 2.78 ± 0.03 is the particle total energy and E o the rest mass. For protons and helium nuclei above rigidities (particle momentum per unit charge [22]) of 100 MV a modulation potential φ, given in units of MV , is such that Φ=|Z|eφ corresponds to the average energy loss from the interstellar medium to a distance r from the Sun.

J(r, E, t) E 2 -E 2 0 = J(∞, E + Φ) (E + Φ) 2 -E 2 0 (1)
The modulation potential can be correlated with both neutron monitor countrate [START_REF] Golden | Proc. 24th Int. Cosmic Ray Conf[END_REF] and solar spot number [START_REF] Wiedenbeck | Proc. 29th Int. Cosmic Ray Conf[END_REF]. At the moment, the expected number of minimum, average and maximum average annual solar spots in 2012 are 42.03, 69.53 and 97.03, respectively. Very similar to 2003 (40.18, 67.18, 94.18 respectively; [6]). The estimated solar modulation parameter in 2003 was 959 MV [START_REF] Alanko-Huotari | [END_REF]. Moreover, the proton data trend observed by the BESS (Balloon-borne Experiment with a Superconducting Spectrometer [26]) experiment in 2002 and 2004 indicated a solar modulation parameter for protons of 1109 MV and 764 MV , respectively (see fig. 2). On the basis of these two last evidences we set to 950 MV the solar modulation parameter as a lower limit for the second half of 2012.

We use the model by Gleeson and Axford to estimate the particle energy spectra during a positive polarity epoch (continuous lines in figs.2 and 3 for protons and helium nuclei, respectively. References to data are reported in [START_REF] Grimani | Proc. 30th Int. Cosmic Ray Conf[END_REF]).

Flux interpolation at the interstellar medium was gathered from [27] and reported in eq. ( 2). The parameters for proton and helium fluxes appear in Table 1. With β and R we indicate particle velocity and rigidity, respectively. The energy spectra during a negative polarity epoch were estimated according to [START_REF] Grimani | Proc. 30th Int. Cosmic Ray Conf[END_REF] for near-solar-maximum conditions (dashed lines in figs. 2 and3). Presently, the Sun has a negative polarity. As pointed out in the previous Section, a change of polarity from -to +, is plausibly expected in 2013. Since at solar maximum a minor effect of solar polarity is observed on cosmic rays [28], we consider our estimated fluxes a lower limit at the time of LISA-PF.

J(∞, β, R) = Aβ P 1 R -P 2 (2)
The particle differential flux interpolation function at 1 AU we use is [29]:

F (E) = A(E + B) -α E γ P articles/(m 2 srsGeV ). (3)
Eq. 3 parameters for proton and helium nucleus flux interpolation are reported in Table 2.

Galactic cosmic-ray short-term variations and fluctuations

27-day variations

The Sun is a massive sphere of plasma and gas rotating at different rates depending on the heliolatitudes. The equator and near equatorial regions of the Sun rotate with a period of about 25-26 days: the Sun's sideral rotation period. For an observer at the Earth this periodicity equals about 27-28 days due to the orbital motion of the Earth. This is called the Sun's sinodic period of rotation. At the poles the Sun rotates with a period of about 36 days. The distribution of active regions and coronal holes on the Sun generates a solar wind asymmetric velocity distribution with respect to heliolongitude and heliolatitude. Since the Sun rotation depends on the reference system of the observer, these recurrent variations are called 27-day variations of GCR intensity [30]. Alania, Gil and Modzelewska [31] showed that the larger amplitudes of the 27-day variations of the GCR intensity and anisotropy are observed during periods of minimum solar activity and positive polarity. In particular, they found that the amplitudes of the 27-day intensity variations present a power-law spectrum in rigidity (A 27 = bR -δ ). This spectrum appears hard (δ=0.54±0.11) during positive polarity periods and soft (δ=0.95±0.12) during negative polarity epochs. From the data reported by Alania, Gil and Modzelewska we have inferred that A 27 varies between 6% at 0.445 GV (0.1 GeV for protons) and 1.2% at 10 GV at solar minimum during positive polarity epochs while goes down to 4% at 0.445 GV and 0.2% at 10 GV during negative polarity periods. GCR anisotropy is smaller than 0.06% (0.03%) during positive (negative) polarity periods.

The 27-day variations were not found to depend on the tilt angles of the heliospheric current sheets according to neutron monitor data.

Forbush decreases

Forbush decreases were discovered by Forbush in 1937. A Forbush decrease is a worldwide drop of the observed GCR intensity occurring within tens of minutes to hours followed by a gradual recovery to the previous average intensity within many hours or days. Forbush decreases can be divided in two classes: sporadic (transient) and recurrent. The sporadic Forbush decreases generate GCR intensity drop lasting 1-2 days then present gradual recovery in, on average, 5-10 days (an example is reported in fig. 4; [32]). The sporadic Forbush decreases of the GCR intensity are associated with major solar flares. These Forbush decreases can be caused by (1) shock and ejecta, (2) shock only or (3) ejecta only [33]. Sporadic Forbush decreases generate drops of the order of 5-10% at 10 GV rigidity. The recurrent Forbush effects (with amplitudes < 3-4% at the rigidity of 10 GV) are associated with the corotating interaction regions in the interplanetary space. Recurrent Forbush effect has approximately symmetric time-profile: the GCR intensity decreases gradually during 5-7 days and recoveries approximately last the same time.

Forbush effects on the 27-day variation of the GCR intensity are unknown up to present.

Forbush decreases affect high energy GCR more than all other shortterm variations and fluctuations. A detailed study of the LISA-PF test-mass charging variation during a Forbush decrease is in preparation. GCR energy differential flux measurements during Forbush decreases were carried out, for example, by the SMILI [START_REF] Clem | Proc. 23th Int. Cosmic Ray Conf[END_REF], MASS89 [START_REF] Grimani | [END_REF] and PAMELA [START_REF] Casolino | [END_REF] experiments. SMILI and MASS89 are balloon-borne experiments flown from Saskatchwan (Canada) soon before (September 1st) and during (September 5th) the Forbush decrease dated September 4th 1989, respectively. Unfortunately, even if these last two experiments show a low energy modulation of the helium nucleus spectrum due to the Forbush decrease, evident normalization problems between the two experiments do not actually give us the possibility to study accurately the energy dependence of the decrease (see fig. 5). This was not the case for the event dated 15 December 2006. The dynamics of the whole event was observed by PAMELA. This satellite experiment, launched on June 15th 2006, is devoted to antimatter search in cosmic rays. Preliminary results were presented by the PAMELA collaboration at the last August European Cosmic-Ray Symposium in Turku (Finland) [START_REF] Casolino | Europ. Cosmic Ray Symp[END_REF]. In fig. 4 the Moscow neutron monitor counting rate represents the trend of the GCR intensity in December 2006. In fig. 6 we have reported the variations of GCR proton flux in different energy intervals (0.4-1 GeV , 1-5 GeV , 5-20 GeV ) during the same period as measured by PAMELA. It appears that the Forbush decrease is softened by solar particles below 1 GeV while an intensity drop of up to 30% is observed between 1 and 5 GeV . Some reduction of the flux is found above 10 GeV .

Hourly and daily GCR fluctuations

Short-term GCR fluctuations observed near Earth are affected by solarterrestrial relations. However, the balloon-borne BESS-Polar I experiment, flown from Antarctica from December 13th through December 21st 2004 [START_REF] Orito | Proc. 31st Int. Cosmic Ray Conf[END_REF], detected galatic proton differential flux variations correlated with the solar activity. Shortly before the BESS-Polar I flight an interplanetary coronal mass ejection or a magnetic cloud reached Earth and on December 14th the count rate of the Bartol South Pole neutron monitor started to recover gradually. The GCR intensity recovery was softened by a high speed stream in the solar wind reaching Earth between December 16th and December 17th. Short-term GCR variations depend on the characteristics of each interplanetary process, however the BESS-Polar I data provide precious clues on these intensity variations as a function of the energy. The proton flux was measured by BESS-Polar I experiment in 4-hour time intervals normalized to the flight average flux in the energy ranges 0.29-0.54 GeV , 0.54-1 GeV , 1-3.4 GeV and 3.4-10 GeV . While diurnal variations do not allow us to give any meaningful explanation to the fluctuations of individual data points [START_REF] Orito | Proc. 31st Int. Cosmic Ray Conf[END_REF], a continuous recovery of the low energy differential flux was observed in agreement with the Bartol South Pole neutron monitor trend modulated by the Sun activity. In particular, between 200 MeV and 1 GeV the proton spectrum varies between -5% to +3% with respect to the average value during the whole flight. Between 1 and 10 GeV the variations appear reduced and consistent with statistical uncertainties [START_REF] Hams | Proc. 31st Int. Cosmic Ray Conf[END_REF].

4 Updated estimate of the number of solar events at the time of LISA-PF Nymmik [7]- [START_REF] Nymmik | Proc. 26th Int. Cosmic Ray Conf[END_REF] has found that the SEP fluence distribution follows a powerlaw trend with an exponential decrease for large fluences. This model applies to solar proton fluences ranging between 10 6 and 10 11 protons cm -2 for particle energies above 30 MeV . The Nymmik results were inferred from the analysis of the spacecraft IMP-7 and 8 measurements of SEP events during the solar cycles 20-22 and from proton fluxes estimated on the basis of radionuclide observations in lunar rocks generated in the last few million years. The Nymmik's model offers the possibility to predict solar events in terms of energy range and of particle peak fluxes instead of fluence only (for a review of other models see [START_REF] Storini | [END_REF]- [START_REF] Lantos | [END_REF] and references therein).

The number of SEP events in individual intervals of fluence during the six months of the expected LISA-PF data taking were estimated according to the March 2010 projections of the number of solar spots reported in fig. 1 (see [20] for details).

We have found a minimum, average and maximum number of solar events in 2012 of 2.92, 4.83 and 6.73, respectively. Half of these are expected in six months. The number of events estimated to occur during the LISA-PF mission per interval of fluence appear in fig. 7.

Conclusions

The most recent projections of the solar cycle 24 allow us to estimate the GCR fluxes at the time of future space missions. In addition to long-term variations, short-term fluctuations were considered. In particular, we found that Forbush decreases generate intensity drops up to 30% in individual energy intervals. Other short-term variations cause maximum variations of a few % at most. We have also estimated the number of solar events with fluences larger than 10 6 protons cm -2 expected during the second half of 2012. This number ranges between 1.5 and 3.4, well below previous expectations. This might mean that no events or, at least, no events with fluences equal or larger than 10 7 protons cm -2 might occur at the time of LISA-PF data taking. We point out that 10 7 protons cm -2 is the intensity of events generating a noise larger than the whole LISA budget at low frequencies (at the peak).

Figure 1 :

 1 Figure 1: Predictions of the average annual sunspot number for the solar cycle 24: 2008 [10] and updated [5]-[6] projections are compared.

Figure 2 :

 2 Figure 2: Estimated GCR proton energy spectra at the time of LISA-PF (2012).

Figure 3 :

 3 Figure 3: Same as fig.2 for helium nuclei.

Figure 4 :

 4 Figure 4: Moscow neutron monitor counting rate in December 2006 [32]. A Forbush decrease occurred on December 15th. We recall that the neutron monitor countrate (Y-axis label) is expressed in neutron impulses per minute.

Figure 5 :

 5 Figure 5: Helium flux measurements carried out by the SMILI and MASS89 experiments on September 1st and 5th 1989, respectively. A Forbush decrease started on September 4th. Normalization problems can be observed between the two experiments since above 1 GeV the SMILI flux assumes smaller values with respect to those of MASS89.

Figure 6 :

 6 Figure 6: GCR integral proton flux observed by PAMELA in various energy intervals between December 12th and December 28th 2006 [37].

Figure 7 :

 7 Figure 7: Estimated number of solar events per interval of fluence at the time of LISA-PF data taking.

Table 2 :

 2 Proton and helium flux parameterization at 1 AU at the time of LISA-PF assuming a positive solar polarity.

	Particle fluxes	A	B	α	γ
	p	18000 1.50 3.90 1.10
	He	850 1.30 3.23 0.48

Acknowledgements

CG and MF acknowledge financial support by the Italian Space Agency within the Program "Study exploration of the Solar System".


Pau Amaro-Seoane
email: pau.amaro-seoane@aei.mpg.de

Miguel Preto
email: miguelp@ari.uni-heidelberg.de

The impact of realistic models of mass segregation on the event rate of extreme-mass ratio inspirals and cusp re-growth

One of the most interesting sources of gravitational waves (GWs) for LISA is the inspiral of compact objects on to a massive black hole (MBH), commonly referred to as an "extreme-mass ratio inspiral" (EMRI). The small object, typically a stellar black hole (bh), emits significant amounts of GW along each orbit in the detector bandwidth. The slowly, adiabatic inspiral of these sources will allow us to map spacetime around MBHs in detail, as well as to test our current conception of gravitation in the strong regime. The event rate of this kind of source has been addressed many times in the literature and the numbers reported fluctuate by orders of magnitude. On the other hand, recent observations of the Galactic center revealed a dearth of giant stars inside the inner parsec relative to the numbers theoretically expected for a fully relaxed stellar cusp. The possibility of unrelaxed nuclei (or, equivalently, with no or only a very shallow cusp, or core) adds substantial uncertainty to the estimates. Having this timely question in mind, we run a significant number of direct-summation N -body simulations with up to half a million particles to calibrate a much faster orbitaveraged Fokker-Planck code. We show that, under quite generic initial conditions, the time required for the growth of a relaxed, mass segregated stellar cusp is shorter than a Hubble time for MBHs with M • 5 × 10 6 M (i.e. nuclei in the range of LISA). We then investigate the regime of strong mass segregation (SMS) for models with two different stellar mass components. Given the most recent stellar mass normalization for the inner parsec of the Galactic center, SMS has the significant impact of boosting the EMRI rates by a factor of ∼ 10 in comparison to what would result from a 7/4-Bahcall & Wolf cusp resulting in ∼ 250 events per Gyr per Milky Way type galaxy. Such intrinsic rate should translate roughly into ∼ 10 2 -7 × 10 2 sbh's EMRIs detected by LISA over a mission lifetime of 2 or 5 years, respectively), depending on the detailed assumptions regarding LISA detection capabilities.

Introduction

Nowadays it is well-established that a massive dark object, very possibly a massive black hole (MBH) with a mass of about 4 × 10 6 M , is lurking in the centre of the Milky Way (Eisenhauer et al. 2005;Ghez et al. 2005Ghez et al. , 2008;;Gillessen et al. 2009). While there is an emerging consensus about the origin and growth of supermassive black holes (SMBH, with masses about or larger than 10 8 M ) (Ferrarese & Ford 2005;DiMatteo et al. 2008;[START_REF] Volonteri | [END_REF], MBHs with smaller masses such as the one in the Galactic centre remain a (relatively) understudied enigma. One of the keys to understanding the growth and evolution of MBHs in this lower mass range resides in the dynamics of stars in their vicinity. This is the case mainly because relaxation times there are low enough that the surrounding stellar systems should have had enough time-through two-body relaxation alone-to evolve towards a steady-state which is independent of the particular initial conditions at the time of formation. The Galactic center is thought to fulfill such condition. It is the universality of such relaxed stellar nuclei that gives us a crucial predictive power on the expected properties of the MBH environment, on the stellar candidates for close interaction with the central MBH and on the resulting gravitational wave (GW) signatures. If, on the contrary, non-relaxed systems were generic, then one would need to resort to case-by-case modelling of each galactic nucleus.

The ideal probe for these innermost regions of galaxies is the GW radiation that is emitted by stellar bhs and other compact objects that come very close to the MBH. One of the main channels for interaction between stars and a central MBH is the adiabatic, slow inspiral of compact remnants (CR) into the MBH due to the emission of GWs-an EMRI. During such an event, the small body effectively acts as a probe of spacetime close to the MBH as its orbit slowly shrinks due to the energy and angular momentum lost in the form of GW radiation. In case of 10 5 -10 6 M MBHs, after some ∼ 10 4 -10 5 orbits in the LISA band (f orb 10 -4 Hz and a periapsis a few × R Schw , since we only consider sources which are completely embedded in the band, and not bursting sources), the small body eventually merges with the MBH. The information contained in the waves will allow us to determine the parameters of these binary system with an unprecedented accuracy (see for instance Barack and Cutler 2004;Babak et al. 2010), corroborate the existence of MBHs and maybe even provide the first direct detection of an intermediate MBH (in case the primary is ∼ 10 3-4 M ).

LISA will thus scrutinize exactly the mass range about which electromagnetic observational information is currently lacking. In its most general form, the EMRI problem-the astrophysical modelling of event rates and parameters for EMRIsspans many orders of magnitude. From the bulge regions at few×10 pc, where the dynamics is essentially collisionless -but from where single stellar bhs and binaries with CRs originate; down to the parsec scale of the nucleus itself which evolves secularly over (local) relaxation timescales; and then further down to milliparsec scales where relativistic effects start to dominate the evolution. But, however, once a steady state configuration establishes itself in the central parsec region, the EMRI rates are rather expected to depend strongly on the (universal) density distribution of CRs within (in order of magnitude) O(0.01pc) from the hole. This is indeed the region from which these inspiralling sources are expected to originate (Hopman and Alexander 2005). The dynamics in this tiny volume has been rather unexplored until the relevance of EMRIs and sub-parsec observations of the Galactic center have raised its interest. Since then, many authors have devoted a number of works to the analysis of this peculiar regime (Sigurdsson and Rees 1997;Freitag 2003;Alexander and Hopman 2003;Hopman and Alexander 2006).

We discuss in this work the stellar distribution of dense stellar systems around MBHs in the LISA mass range. Realistic modeling of mass segregation-which is the natural outcome for any realistic stellar population-will strongly impact the expected EMRI rates, since it favors the accumulation of heavier objects towards the center (Hopman and Alexander 2006;Alexander and Hopman 2009;Preto and Amaro-Seoane 2010a). In Section 2, we begin by summarizing the results obtained by Preto and Amaro-Seoane (2010a) that show how to calibrate the FP calculations with direct Nbody simulations ‡; then, still in the same section, we present new results concerning the robustness of N -body realisations of stellar cusp growth with respect to the total particle number N . In section 3, we present new results on the growth of stellar cusps from a variety of initial conditions resulting from carving a cavity in the star's phase space distribution function. This is motivated from a variety of astrophysical mechanisms that may lead to cusp destruction; and these mechanisms are critically assessed in the end of the section. We show that, under very generic circumstances, the time required for the growth of a cusp is shorter than a Hubble time. Therefore, quasi-steady, mass segregated, stellar cusps are expected to be common around MBHs in the LISA mass range. But see Merritt (2010) and Madigan et al. (2010) for different perspectives. EMRI detection rates for LISA are expected to peak for M • ∼ 10 5 -10 6 M (Gair 2009) leading us to conclude that at least a sizeable fraction of these events should originate from strongly segregated cusps. Finally, in Section 4 we present new estimates on the expected EMRI rates in mass segregated nuclei and conclude that our realistic modeling of mass segregation has a significant impact on these rates.

Mass segregation

The distribution of stars around a massive black hole is a classical problem in stellar dynamics (Bahcall and Wolf 1976;Lightman and Shapiro 1977). Bahcall and Wolf (1976) have shown, through a kinetic treatment that, within the radius of gravitational influence of the hole r h , in case all stars are of the same mass, this quasi-steady distribution takes the form of power laws, ρ(r) ∼ r -γ , in physical space with ρ(r) the stellar density at a radius r and f (E) ∼ E p in energy space (with E the energy and ‡ Direct N -body simulations compute the gravitational accelerations between particles using the exact Newton's law, without introducing any approximations to compute the gravitational potential (Binney and Tremaine 2008). γ = 7/4 and p = γ -3/2 = 1/4) §. This is the so-called zero-flow solution for which the net flux of stars in energy space is precisely zero. Preto et al. (2004) and Baumgardt et al. (2004) were the first to demonstrate the robustness of the corresponding directsummation N -body realizations, and have therefore validated the assumptions inherent to the Fokker-Planck (FP) approximation-namely, that scattering is dominated by uncorrelated, 2-body encounters and, in particular, dense stellar cusps populated with stars of the same mass are robust against ejection of stars from the cusp. The latter result is not trivial as for a BW γ = 7/4 cusp stellar densities are extremely high at the center and the fraction of stars with speeds close to the escape velocity from the cusp is quite high at all radii r r h , with r h the influence radius of the MBH (Preto 2010).

Single mass models are very poor approximations of real stellar populations. To first order of approximation, an evolved stellar population can be represented by two (well-separated) mass scales: one in the range O(1M ) corresponding to low mass mainsequence stars, white dwarfs (WDs) and neutron stars (NSs); another with O(10M ) representing stellar bhs. Therefore, for simplicity, here we restrict our discussion to models with two mass components and leave the more general case to another work in preparation (Preto and Amaro-Seoane 2010b).

When stars of two different masses are present, there is mass segregation which is a process by which the heavy stars accumulate near the center while the lighter ones float outward [START_REF] Spitzer | Dynamical evolution of globular clusters[END_REF]Khalisi et al. 2007). Accordingly, stars with different mass get distributed with different density profiles. Bahcall and Wolf (1977), henceforth BW77, have argued heuristically that a scaling relation p i = m i /m j × p j (where the subindices i, j refer to the light or heavy components) establishes itself and depends only on the mass ratio. Here, as in the single-mass case, the crucial assumption is that all components are abundant enough that they undergo enough scattering among themselves and with the other components as to stabilize into an approximate zero-flow solution. Obviously, this cannot happen independently of the number fraction of the different stellar masses (Alexander and Hopman 2009;Preto and Amaro-Seoane 2010a). In the realistic situation where the number fraction of heavy objects (in our case, stellar bhs) is small, a new solution coined by Alexander and Hopman (2009) as strong mass segregation (SMS) obtains with density of heavy objects scaling as ρ H (r) ∼ r -α , where α 2. The solution has two branches and can be parametrized by the parameter ∆ = D

(1)

HH + D (2) HH D (1) LH + D (2) LH ≈ N H m 2 H N L m 2 L 4 3 + m H /m L , (1)
where N L and N H are the total number of light and heavy stars, m L and m H are the corresponding individual masses. ∆ provides a measure of the importance of the heavy star's self-coupling relative to the light-heavy coupling (in terms of the 1 st and 2 nd order diffusion coefficients); and it depends essentially on the mass and number ratios, which is one parameter more than proposed by BW77. The weak branch, for ∆ > 1 corresponds to the scaling relations found by BW77; while the strong branch, for ∆ < 1, generalizes the BW77 solution ¶. Stellar populations with continuous star formation and an initial mass function (IMF) given by dN/dM ∝ M -α will be characterized by ∆ < 1 if α 1.8 and ∆ < 1 otherwise; and, in particular, Salpeter and Kroupa's IMF generate evolved stellar populations with ∆ < 1 (Alexander and Hopman 2009).

∆ = 0.08 -4 -2 0 2 4 log 10 ρ(r) -1.5 N=512K M BH =0.01 -2.1 ∆ = 0.08 -4 -2 0 2 4 -4 -3 -2 -1 0 1 r -1.5 N=124K M BH =0.05 -3 -2 -1 0 1 r -1.8 ∆ = 13.2
curves are from NB simulations. The agreement between both methods is quite good. The mass ratio between heavy and light stars is R = 10, representing the expected typical mass ratio between light stars (MS stars, WDs and NSs) and stellar bhs, as explained in the text; the number fraction of heavy stars f H = 2.5 × 10 -3 (top and middle panels) and f H = 0.429 (lower panels), corresponding to the strong and weak segregation regimes respectively. The initial condition is a Dehnen profile with central slope γ = 1 for the top and bottom panels (Preto and Amaro-Seoane 2010a), γ = 1/2 in the middle panel; a central MBH with 5% of the total mass of the cluster and 1% likewise. The particle number is N = 124, 000 (top and bottom) and N = 512, 000 (middle). The asymptotic slope γ H decreases from 2 to ≈ 7/4 when moving from the strong to the weak branch of the solution. The asymptotic slope γ H ≈ 3/2 throughout, or just slightly below this value. The arrows point to radii r h and 0.1r h .

There is a straightforward physical interpretation for the strong branch of mass segregation. In the limit where heavy stars are very scarce, they barely interact with each other and instead sink to the center due to dynamical friction against the sea of light stars. Therefore, a quasi-steady state develops in which the heavy star's current is not nearly zero and thus the BW77 solution does not hold exactly anymore. Indeed, in the limit where the number fraction f H of heavy stars is vanishingly small, as is the case of nuclei with realistic IMFs, the stellar potential is dominated by the light component. In this case, the light stars should evolve as if in isolation and develop a γ L ∼ 7/4 density cusp. The scarce heavy stars sink to the center due to dynamical friction against the background of light stars, and will not exert any significant back-reaction on them (Preto 2010).

Figure 1 displays the FP and NB evolutions of the spatial density ρ L (r) and ρ H (r) for models with two mass components corresponding to different initial profiles, MBH masses and total particle number N . The starting models are either γ = 1 or γ = 1/2 Dehnen profiles for both components with a MBH of 1% or 5% of the total mass of the cluster. Dehnen density profiles are defined by ρ(r) = (3-γ)M T OT /4πr γ (r b +r) 4-γ , have total stellar mass M T OT , an inner (outer) logarithmic slope γ (4 -γ) and a break radius r b = 1 (which is larger than r h in all cases). We adopt units where G = M T OT = 1. The density of both components reaches a quasi-steady state within ∼ 0.2T rlx (r h ), where T rlx (r h ) relaxation time measured at the influence radius (Preto and Amaro-Seoane 2010a). The top and middle panels display the strong mass segregation solution with γ H ∼ 2.1 as expected for ∆ = 0.08 (f H = 2.5 × 10 -3 ); while, in the bottom panel, ∆ = 13.2 (f H = 0.429) displays the weak solution for which γ H ∼ 7/4. The former value was chosen to be close to the number fraction of stellar bhs to be expected from a standard Salpeter or Kroupa's IMF; the latter value is chosen to be representative of the regime of weak segregation studied by BW77. One can see from Figure 1 that in the case of weak segregation ρ H > ρ L everywhere due to the extremely high number of heavy objects; in contrast, in the SMS regime ρ H > ρ L only for r 0.01r h (and the light objects dominate in number almost everywhere). In all cases the asymptotic slopes are valid within ∼ 0.1r h and are totally predictable once ∆ is known. These results agree pretty well with the predictions for the SMS regime (Alexander and Hopman 2009).

The particle number in our direct-summation N -body simulations sample ranges from N = 124, 000 to N = 512, 000; our results do not show evidence of any dependence on total N , nor on the initial value of γ, once the results are re-scaled appropriately (i.e. measured in terms of the relaxation time). The agreement between NB and FP methods is quite good in all cases.

Cusp Re-growth

Current observations: A missing cusp

We have seen that theory predicts a steady state cusp that reaches extremely high densities in the center near the MBH. Furthermore, given a normalization at, say, r h and a knowledge of the stellar mass function (and thus of ∆), the density profile inside r h becomes completely determined. But observations are much more complicated to interpret. First, one must realize that there are very few galaxies for which the influence radius r h can be resolved. In fact, except for the nearest galaxies, r h covers an angular region in the sky which is too small to be resolved even with the HST. The HST has an angular resolution of ∼ 0.1 . In the case of galaxies in the Virgo cluster, at ∼ 17 Mpc of distance, it can only resolve regions with linear sizes 8.25 pc. Therefore, HST would not be able to resolve SgrA*'s radius of influence if it were at the distance of Virgo. Since

r h ∝ M 1/2
• , it can only start to resolve the influence radius of Virgo's MBHs that have masses M • 4 × 10 7 M . Second, even in the few cases for which r h can be resolved to some extent, it still is necessary to assess whether the observed stars (only those that are bright enough to be detected) really trace the underlying (dynamically dominant) invisible population. Third, given the fact that, as we have seen, stars tend to segregate by mass, there is an extra uncertainty related to the unknown stellar mass function. Moreover, there are indications that star formation events are common in galactic nuclei and furthermore that the resulting IMF in these sub-parsec regions may be substantially different from that of the field stars and biased towards heavy masses (Bartko et al. 2010). Finally, it is necessary to deproject the observations and, in the (inevitable) absence of complete knowledge of phase space coordinates, one must rely on kinematic assumptions regarding the (an-)isotropy of stellar velocities and on the three dimensional shape of the stellar system. [0.1T rlx , 0.2T rlx ]-the time for cusp regrowth if there is no hole in the initial DF. The three dashed lines above the shaded region represent the average time needed for the cusp re-growth in case one imposes an initial cavity with size R cav = 0.5, 1 or 2 pc. The horizontal dashed curve represents 13 Gyr. It can be seen that the time needed to re-grow a cusp around MBH with masses M • 5 × 10 6 M is below a Hubble time so long as the initial cavity is smaller than 2 pc.

Nevertheless, it has come as a surprise that very recent spectroscopic observations of the Galactic center revealed a core (or even a dip) in the surface distribution of the old stellar population (essentially red giants) which should have had time to relax into a cuspy density profile [START_REF] Buchholz | [END_REF]Do et al. 2009). The caveat is that the detected stars are still a small fraction, of about 5%, of the stellar population as a whole and therefore do not exclude the presence of an extended dark cluster (presumably made of stellar bhs and other CRs)-which would indeed agree with our theoretical expectations.

Carving a hole in the stellar distribution

To assess the likelihood that the Galactic center is indeed unrelaxed, it is natural to ask: how long does it take to re-grow a cusp if, at some point, it has been destroyed? A complete answer will, of course, depend on the extent to which the cusp was destroyed, i.e. how much mass was expelled from the original cusp and over which radial range. At this level, it does not matter very much which mechanism led to the destruction of the cusp. We discuss briefly possible scenarios for cusp destruction at the end of the section.

In order to investigate this question, we have concocted a set of initial conditions purported to mimic the outcome of a destroying cusp event-such as the carving of a cavity in phase space through the ejection of stars by, say, an infalling IMBH or, following a major merger, by a MBH. We model the outcome of such an event by imposing that all stars with binding energies larger than some E 0 or almost equivalently, with semimajor axis smaller than GM • /2E 0 , are not present in the initial DF. In fact, inside r h the MBH dominates the gravitational potential and E ∼ GM • /2a. We thus set up an initial Dehnen model with f (E) = 0 for E > E 0 -in other words, there is an initial cavity in the phase space DF, but not in physical space as the stars with lower energy still entail ρ(r) ∼ r -1/2 at the center, although with a smaller amplitude than the original model. The values of E 0 were chosen to represent cavities of size R c = 0.5, 1, 2 pc resulting from the inspiral of an IMBH/MBH. Note that these models are, by construction, isotropic in the velocity distribution. + Our fiducial model is a Milky Way type nucleus with M • = 4 × 10 6 M , some 10 6 M in total stellar mass inside 1 pc distributed according to an initial central density slope γ = 1/2, two components with masses m L = 1M and m H = 10M , and 0.1% of stellar bhs by number. When the stellar distribution has no phase space cavity, this translates into having stars down to roughly 10 -5 pc. Having validated the FP models with detailed N -body simulations, we now proceed in the rest of the paper to describe results obtained with the (much faster) FP approach.

Figure 2 shows the times for cusp re-growth computed with FP for different galactic nuclei models. The time for cusp re-growth is defined as the time it takes for a given initial density profile (ρ(r) in space or f (E) in phase space, with or without an initial cavity) to reach its asymptotic slope, which depends on ∆, down to r ∼ 0.01r h . This is indeed the scale which is resolved by recent observations of the Galactic center (Schödel et al. 2009). The shaded region represents the time of cusp re-growth for a range of R and f H (all in the SMS regime, ∆ < 1) for the case where f (E) extends to high E without any cut. It can be seen that, for M • 10 7 M , cusps grow in less than a Hubble time; in the particular case of the Milky Way nucleus with M • ∼ 4 × 10 6 M , it takes no longer than ∼ 4.8 Gyr to fully re-grow a steady-state, mass segregated, stellar cusp and only ∼ 2.4 Gyr to have it grown down to 0.01r h . If, instead, an initial cavity is imposed at the center with size R cav = 0.5, 1 or 2 pc in case of the Milky Way (or R cav = 0.2r h , 0.4r h or 0.8r h in case of a generic nucleus), times for re-growth are represented by the dashed curves above the shaded region. In this case, times for cusp re-growth increase; in the Milky Way case, it becomes ∼ 4.8, 7.2 or 12 Gyr, respectively. Note that, in the mass range 10 5 -10 6 M , the times for cusp re-growth are definitely much shorter than a Hubble time-even if a fairly large cavity of size comparablee to r h is hypothesized. The full curve represents the relaxation times computed at the radius + We assume that the timescale for isotropization of velocities is much shorter than that associated with the cusp re-growth; in any event this should not affect our estimates by more than 10% or 20% maximum.

of influence r h , while the dashed curves represent the actual times for cusp re-growth as measured from FP calculations (Preto and Amaro-Seoane 2010a).

It is difficult to devise plausible mechanisms for the formation of such large cores in the stellar distribution. For instance, the inspiral of an IMBH of mass M • ∼ 10 3-4 M that forms an unequal-mass binary with the MBH and ejects stars through three body encounters would tend to progressively wipe out the stellar cusp. However, the core radius carved by such an event is r c ∼ 0.02 -0.04 pc (Baumgardt et al. 2006) and thus a steady inflow of such IMBHs (one every 10 7 years for a Hubble time) would be required in order to carve a large core 50 or 100 larger. Such large inflow of IMBHs have been proposed by Portegies Zwart et al. (2006). This does not seem very likely anymore in light of the fact that such IMBHs were hypothesized to be formed by runaway mergers of stars in the center of globular clusters. However, at solar metallicities, such mechanism seems very inneficient. Mass loss due to very strong winds severely limits the growth of the stellar object being formed and the likely end result of a runaway merger is a ∼ 100M Wolf-Rayet star. At lower metallicities, mass loss is lower and the remnant can be more massive ∼ 260M , but in any case it will not form an IMBH (Glebbek et al. 2009). In sum, it looks very unlikely that sufficient IMBHs can be formed in order to generate such steady inflow to the Galactic center. Another possibility would be that SgrA* is a binary MBH, but this would most likely imply that there has been a, more or less recent, major merger involving the Milky Way. This would contradict the apparent pure-disk nature of the Galaxy, as theoretical interpretations of stellar kinematic data of the Galactic Bulge seem to favor that the Bulge is part of the disk and not a separate component resulting from a merger Shen et al. (2010)y-aside from the fact that there are strong constraints from the SgrA* proper motion (Reid and Brunthaler 2004).

Stars in a Keplerian potential, GM • /r, do not precess because of the 1 : 1 resonance between their radial and azimuthal frequencies. Resonant relaxation (RR) results from the coherent torques that such stars exert on each other leading to a fast evolution of their orbital angular momenta over timescales T pr , where T pr is the precession timescale due to departures from an exact Kepler potential (Rauch and Tremaine 1996). Madigan et al. (2010) suggest that RR, by increasing the rate of angular momentum diffusion in the near-Keplerian gravitational potential around the MBH, may boost the tidal disruption rate of stars and could thus create a near-cavity (out to ∼ 0.1 pc) in the stellar distribution. It is certain that RR operates to some extent in the inner parsec, but we doubt it can completetly explain the dearth of red giant stars there or, more generally, the full destruction of a cusp-including CRs such as stellar bhs * First, their final density distribution does not show a cavity, nor a shallow cusp profile, for r 0.1 pc; instead, they get final slopes γ ∼ 1.5 for r 0.1 pc. This is in contrast with observations which show a decaying density for r 0.24 pc [START_REF] Buchholz | [END_REF]. Second, we believe that their Monte-Carlo calculations severely underestimate the rate at which the cusp re-grows; in fact, they obtain a timescale which is ∼ 10 -30 times longer than that obtained with our N -body simulations (which are free from any simplifying dynamical assumptions), and also from ours and their own recent FP calculations (Hopman and Madigan 2010). Third, were they to use the latter rates, and given that the time RR takes to deplete the cusp is of the same order as the time we obtain for cusp re-growth, the net effect of RR on the cusp would likely be minute. Moreover, stellar bhs cannot be tidally disrupted, and this makes them less susceptible to be extracted from the cusp than 10M stars.

EMRIs rates

Adiabatic and abrupt EMRIs: Estimation of the rates

Given a steady state stellar bhs continue to diffuse in (E, J)-space and some of them eventually come into close interaction with the MBH. During a close interaction, a stellar bh can either be promptly scattered into the MBH, accompanied by a single or a few brief bursts of GWs in the LISA band-the so-called "direct-plunges"-, though they are not likely detectable unless if emitted from the Galactic center (Hopman et al. 2007), or scattered outwards in the cusp. In either case, it does not live enough to become an EMRI. Alternatively, it may undergo a very slow, adiabatic, inspiral without being appreciably disturbed by other stars and, in this case, it will eventually become an EMRI detectable by LISA. An EMRI object thus has to spend very many orbits without being significantly scattered by the gravitational tugs of the other stars. In other words, they must fullfill the following inspiralling criterion: the time T GW it takes for the inspiral, due to orbital energy lost by GW emission only, must be shorter than the typical time T J it takes on average to drift in angular momentum by an amount J which equals its orbital angular momentum. Otherwise, they will be promptly captured by the MBH before entering the LISA band. The inspiral criterion can be stated in terms of the parameter s being smaller than unity, s = T GW /T J < 1. For T GW > T J , it is almost certain that this object has either taken an almost radial orbit and fallen into the MBH as a direct plunge or has been scattered outwards.

It turns out that this parameter simply scales with orbital's semimajor axis: s ∝ a 3/2-p (Hopman and Alexander 2005), which means that it is a decreasing function of a so long as p < 3/2. This is indeed the case in both regimes of mass segregation. Furthermore, Hopman and Alexander (2005) have shown that the probability for a successful inspiral as a function of orbital semimajor axis (or energy) is almost a step function of semimajor axis. If a < a GW , it is almost certain that the stellar bh will become an EMRI; it will almost certainly not become one in case the inequality sign is reversed (and the width of the "transition region" is very small). This crucial threshold quantity demarcates the orbits which are close enough to the MBH to sucessfully decouple from the rest of the cluster and undergo the slow, adiabatic inspiral that defines an EMRI from those more weakly In steady state, on average each star that drifts outward by an amount J will be compensated by another that drifts inward by the same amount. This balance only breaks down for those orbits that fall on to the hole, since there are obviously no stars coming out of it to keep detailed balancing.

bound orbits that will be perturbed out of the EMRI tracks due to scattering with other stars.

Therefore, in order to estimate the EMRI event rate, given a steady state f (E), obtained via the FP equation, one essentially counts the number of stars that populate the region of phase space for which the inspiralling criterion above is satisfied and divide it by the local relaxation time. Note that here, for simplicity, we ignore other driving mechanisms-in particular, we ignore resonant relaxation. Under these assumptions, the EMRI rate for stellar bhs is approximately given by

Γ EMRI = f • +∞ E GW dE n(E) ln(J c (E)/J lc ) T rlx (E) , (2)
where f • is the number fraction of SBHs in the stellar population, n(E) is the total number of stars per unit energy(n(E) ∝ f (E), see Preto ( 2010)), J c (E) = √ GM • /2E is the specific angular momentum of a circular orbit of energy E, J lc = 4 GM • /c is the losscone angular momentum and T rlx = 0.34

σ 3 /[G 2 (m • ρ • + m * ρ * ) ln Λ]
is the relaxation time. The log term in the denominator in (2) arises from the phase space (partial) depletion resulting from the presence of the loss cone. The conversion between r and E is, for r r h , E(r) = GM • /2r or E = GM • /2a. The critical radius a GW , or energy E GW , for EMRIs is approximately a GW = 0.01r h ; and, to first order, a GW is independent of M • (Hopman and Alexander 2005).

The relevance of realistic models of mass segregation for the rates

The weak regime of SMS, and corresponding BW solution, would lead to a fairly high intrinsic rate, per galaxy, of EMRIs. In fact, Figure 3 shows that, for a Milky Way nucleus, in case ∆ > 1, the intrinsic EMRI rate is 10 3 per Gyr. This is, however, unrealistic as such scenario pressuposes an unrealistically high number fraction of bhs (f • 0.0325 for ∆ > 1). In the more realistic case, when ∆ ∼ 0.03 the BW solution would entail a strong supression of the EMRI rate to-at best-a few tens of events per Gyr. This is where SMS solution appears to rescue us. SMS implies a higher density of bhs inside r h as compared with the γ = 7/4 solution, and in this way-by decreasing the local T rlx and increasing n(E) close to the MBH-it partially, but not completely, compensates for the small number fraction of bhs entailed by realistic mass functions. as a function of the parameter ∆. This is computed from a two-component mass segregated stellar cusp (γ H ≈ 2.1 and γ L ≈ 1.5) with mass ratios R = 10 and 15 obtained from FP calculations. In the case of the fiducial value f • = 10 -3 , ∆ ≈ 0.03; in those circumstances, each Milky Way like nucleus will produce on average ∼ 250 stellar bh EMRIs per Gyr.

In order to quantitatively evaluate the boost Γ SMS /Γ BW to the EMRI rates from SMS, for a given ∆ and a fixed mass normalization at r h , one needs to estimate what would be the rate if the spatial and phase space densities were determined by the γ = 7/4 cusp for r 0.1r h . This is done as follows: we define analytically both ρ(r) and f (E) that would result from a γ = 7/4 inside 0.1r h ρ

(r) = ρ F P (r), r>r L ρ(r) = ρ F P (r L ) × r L r 7/4 , r ≤ r L , (3)
and

f (E) = f F P (E), E<E L f (E) = f F P (E L ) × E E L 1/4 , E ≥ E L , (4)
where the indices FP mean that the profile is taken from the Fokker-Planck calculation. r L (and E L ) is a reference radius (energy) chosen according to r L ∼ 0.1r h . Figure 4. Boost on the EMRI rates from strong segregation. One can see that, for values of ∆ < 1, there is a significant boost to the EMRI rates in comparison to which it would result in the case of a γ = 7/4 BW cusp. In particular, for our fiducial value ∆ ∼ 0.03 (f • ∼ 10 -3 ), the boost is of order of a factor 10 with respect to a 7/4-BW cusp with the same mass normalization at r = 1 pc.

Figure 4 shows the boost to the EMRI rates due to SMS relative to what would be obtained from a BW profile.

Going from an unrealistically high f • , as adopted by BW77 (say ∆ = 3), to a more realistic f • (say ∆ = 0.03), while neglecting the existence of SMS, one supresses the EMRI rate by factors of ∼ 100 -150 (the former would lead to ∼ few × 10 3 EMRIs per Gyr; the latter is reduced to ∼ few tens per Gyr). However, by taking into account the SMS solution, for this low ∆ = 0.03, we boost again the rates by a factor close to 10, thus partially compensating the reduction of EMRIs (from few tens to a few hundred per Gyr; in fact, there are ∼ 250 per Gyr in case ∆ = 0.03 for a Milky Way nucleus). We conclude that the apparently inocuous and tiny change of the logarithmic slope from γ H = 7/4 to γ H ∼ 2 can have a substantial effect (a factor of ∼ 10) on the expected EMRI rate.

Figure 5. EMRI rates as a function of MBH mass in strongly segregated nuclei. )

log 10 M BH (in M O • ) -0.19 -0.19 R=10, f BH =10 -3 R=10, f BH =10 -2 R=15, f BH =10 -3
The EMRI rate depends on the MBH mass, Γ SMS ∝ M -0.19 • . Shown are curves for ∆ = 0.03 (f • = 10 -3 and 10 -2 ) and three different mass ratios between heavy and light stars, R = 10 and 15.

Figure 5 shows the dependence of the intrinsic EMRI rate on the mass of the central MBH, where the validity of the M • -σ relation was assumed (Ferrarese & Ford 2005). We see that the EMRI rate for stellar bhs scales as ∝ M -0.19 • , independently of R and f • . Its absolute normalization depends obviously on the number fraction f • of sbhs, in agreement with Figures 3 and4.

One can make a rough conversion of the estimated intrinsic rate into LISA detection rates. Following Gair (2009), who made a number of assumptions regarding the local density of MBHs and its spin distribution, plus on the LISA detection capabilities, we find that according to its equation ( 7), LISA will see around ∼ 10 2 -7 × 10 2 EMRI events during a 2-year or 5-year mission, respectively. Note that these rates may change by factors of ∼ 2-3 as a function of corrections to the local MBH mass density (Graham and Driver 2007); moreover, the uncertainties regarding the efficiency of RR and other channels may still affect the rate predictions by one or two orders of magnitude. A lot of work still remains to be done; nevertheless, the consequences regarding the SMS regime are significant and under control.

Conclusions

We have considered simplified stellar models of galactic nuclei, with only two mass components, which harbor MBHs that fall into the LISA detection bandwidth. For quite generic initial conditions, such stellar clusters are expected to have reached a relaxed, mass segregated, steady state which is independent of initial conditions at the time of formation. Strong (realistic) mass segregation is a robust outcome from the growth and evolution of stellar cusps around MBHs in the mass range 10 4 -10 7 M to which LISA will be sensitive. Our N-body results validate the Fokker-Planck description of the bulk properties of the stellar distribution. SMS boosts the EMRI event rates with respect to what would be implied by a shallower stellar density profile (e.g. γ = 7/4, which has been the working assumption of almost all event rate estimates in the literature so far) that also respect the mass normalization obtained from observations of the Galactic center at 1 pc from the hole. In particular, our fiducial models of the Galactic center are enhanced by a factor of ∼ 10-leading to a predicted value of ∼ 250 stellar bh EMRIs per Gyr. The FP formalism assumes two-body relaxation as the only dynamical driver present-this could be a severe restriction at radii 0.01r h , inside which even the NB simulations with higher N in our sample start to run out of particles and where RR could play an important role (Hopman and Alexander 2006;Madigan et al. 2010). Other crucial mechanisms are resonant relaxation, (small) triaxiality of the galactic potential, tidal separation of binaries and massive perturbers (see e.g. Amaro-Seoane et al. 2007, for a review). These are the subject of our current research work, and the extent to which they can significantly affect the EMRI rates is still an open question.

Figure 1 .

 1 Evolution of density profiles. Mass density profiles, ρ L (r) (left panels) and ρ H (r) (right panels) at the end of the integrations, after ≈ 0.2T rlx (r h ). Red curves are from FP calculations, green

Figure 2 .

 2 Figure 2. Time for cusp re-growth. Single-mass relaxation time at r h for singlemass cored models as a function of MBH mass. The shaded area covers

Figure 3 .

 3 Figure 3. EMRI rate as a function of ∆. The number of stellar bh EMRI events per Gyr in a Milky Way type nucleus (M • = 4 × 10 6 M and M * (< 1pc) = 10 6 M )

§ We note that 12 years before the work of BW,Gurevich (1964) derived a similar solution for how electrons distribute around a positively charged Coulomb center, which is the equivalent of the MBH in our case.In this work, a nucleus is said to be a core if γ < 1; it is said to be a cusp if γ > 1.

¶ The choice of the names is based upon the resulting slopes in the density profiles, which are steeper (stronger) or shallower(weaker)

† † GRACE: see http://www.ari.uni-heidelberg.de/grace

Acknowledgments

It is a pleasure for PAS to thank Tom Prince for the invitation to give a plenary talk in the symposium. He is equally indebted with Esmeralda Martínez for discussions as well as with Vivian J. Drew for her great sense of humour and coordination of the workshop. MP thanks Lauren B. for interesting comments. PAS and MP are partially supported by DLR (Deustches Zentrum für Luft-und Raumfahrt). The simulations have been carried out on the dedicated high-performance GRAPE-6A clusters at the Astronomisches Rechen-Institut in Heidelberg † †, which was funded by project I/80 041-043 of the Volkswagen Foundation and by the Ministry of Science, Research and the Arts of , an also at the Tuffstein cluster of the AEI.