RAG.ipynb

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      "source": [
        "!pip install -qU \\\n",
        "  transformers==4.31.0 \\\n",
        "  sentence-transformers==2.2.2 \\\n",
        "  pinecone-client==2.2.2 \\\n",
        "  datasets==2.14.0 \\\n",
        "  accelerate==0.21.0 \\\n",
        "  einops==0.6.1 \\\n",
        "  langchain==0.0.240 \\\n",
        "  xformers==0.0.20 \\\n",
        "  bitsandbytes==0.41.0 \\\n",
        "  pypdf==3.17.4 \\\n",
        "  pinecone-client==2.2.2"
      ]
    },
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      "source": [
        "\n",
        "from torch import cuda\n",
        "from langchain.embeddings.huggingface import HuggingFaceEmbeddings\n",
        "\n",
        "embed_model_id = 'intfloat/e5-base-v2'\n",
        "\n",
        "device = f'cuda:{cuda.current_device()}' if cuda.is_available() else 'cpu'\n",
        "\n",
        "embed_model = HuggingFaceEmbeddings(\n",
        "    model_name=embed_model_id,\n",
        "    model_kwargs={'device': device},\n",
        "    encode_kwargs={'device': device, 'batch_size': 32}\n",
        ")"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "6TYTU7gJ186X",
        "outputId": "11d5541c-ff9f-4f3d-fba2-3d554b83d9d5"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "We have 2 doc embeddings, each with a dimensionality of 768.\n"
          ]
        }
      ],
      "source": [
        "\n",
        "docs = [\n",
        "    \"this is one document\",\n",
        "    \"and another document\"\n",
        "]\n",
        "\n",
        "embeddings = embed_model.embed_documents(docs)\n",
        "\n",
        "print(f\"We have {len(embeddings)} doc embeddings, each with \"\n",
        "      f\"a dimensionality of {len(embeddings[0])}.\")\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "metadata": {
        "id": "0K9tLnxk7I7k"
      },
      "outputs": [],
      "source": [
        "from langchain.document_loaders import PyPDFDirectoryLoader\n",
        "from langchain.text_splitter import RecursiveCharacterTextSplitter"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 5,
      "metadata": {
        "id": "HhfpNkii7U6c"
      },
      "outputs": [],
      "source": [
        "def read_doc(directory):\n",
        "    file_loader=PyPDFDirectoryLoader(directory)\n",
        "    documents=file_loader.load()\n",
        "    return documents"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 6,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "fIfmZu3V7xHu",
        "outputId": "689b27da-7190-4585-cbce-5d8861dc02f5"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "[Document(page_content='DRAFT VERSION JANUARY 1, 2024\\nTypeset using L ATEXtwocolumn style in AASTeX63\\nBinary mergers in the centers of galaxies: synergy between stellar flybys and tidal fields\\nMILAWINTER -GRANI ´C,1CRISTOBAL PETROVICH ,1, 2AND VALENTÍN PEÑA-DONAIRE1\\n1Instituto de Astrofísica, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, 782-0436 Macul, Santiago, Chile\\n2Millennium Institute of Astrophysics MAS, Nuncio Monseñor Sótero Sanz 100, Of. 104, 750-0000 Providencia, Santiago, Chile\\nABSTRACT\\nGalactic centers are very dense and dynamically active environments, often harboring a nuclear star cluster and\\nsupermassive black hole at their cores. Binaries in these environments are subject to strong tidal fields that can\\nefficiently torque its orbit, exciting near unity eccentricities that ultimately lead to their merger. In turn, the\\nfrequent close interactions due to passing stars impulsively perturb the orbit of the binary, generally softening\\ntheir orbit until their evaporation, thus potentially hindering the role of the tidal fields to drive these mergers.\\nIn this work, we study the evolution of compact object binaries in the galactic center and their merger rates,\\nfocusing for the first time on the combined effect of the cluster’s tidal field and flyby interactions. We find a\\nsignificant synergy between both evolution processes, where merger rates increase by a factor of ∼10−30\\ncompared with models in which only flybys or tidal fields are taken into account. This synergy is mainly a\\nconsequence of the persistent tides-driven eccentricity excitation that is enhanced by the gradual diffusion of\\njz(z-component of the angular momentum) driven by flybys. The merger efficiency peaks when the diffusion\\nrate is ∼10−100times slower than the torquing due to the tidal field. Added to this synergy, we also find that\\nthe gradual softening of the binary can lift the relativistic quenching of initially tight binaries, otherwise unable\\nto reach extreme eccentricities, and thus expanding the available phase-space for mergers. Cumulatively, we\\nconclude that despite the gradual softening of binaries due to stellar flybys, these greatly enhance their merger\\nrates in the centers of galaxies by promoting the eccentricity excitation driven by the tidal fields.\\nKeywords: galactic center – compact objects – star clusters – supermassive black holes – stellar dynamics\\n1.INTRODUCTION\\nMost nearby galaxies are known to harbour very dense\\nstellar clusters at their core (Neumayer et al. 2011; Turner\\net al. 2012; Georgiev & Böker 2014). These nuclear star clus-\\nters (NSCs) tend to have masses of roughly 105−108M⊙, a\\nsmall effective radii of only a few parsecs, and usually have\\na supermassive black hole (SMBH) residing at their center.\\nThis is the case for our own galaxy, containing an NSC with\\na mass of ∼3×107M⊙(Schödel et al. 2014, 2018) and\\nthe central SMBH SgrA∗with a mass of about 4×106M⊙\\n(Ghez et al. 2005; Gillessen et al. 2009). This makes the\\ngalactic center (GC) a very dense and dynamical environ-\\nment, which turns it into an ideal laboratory to test several\\nastrophysical phenomena involving stellar dynamics, general\\nrelativity (GR), and gravitational waves (GWs).\\nIn the last decade there has been an increasing interest in\\ngravitational wave astrophysics, due to GW detections from\\nthe LIGO and VIRGO interferometers. Compact object bi-\\nnary mergers are known to be one of the main GW emitting\\nevents (Abbott et al. 2016, 2017), therefore recent studies\\nCorresponding author: Mila Winter-Grani ´c\\nmila.winter@uc.clhave been focused on the dynamical mechanisms that lead\\nto these mergers in order to better understand GW sources.\\nParticularly in GCs, one of the most important physical pro-\\ncesses leading up to mergers is the torquing of binary’s orbits\\nproduced by the SMBH, process known as the von Zeipel-\\nLidov-Kozai (ZKL) mechanism (Antonini & Perets 2012;\\nStephan et al. 2019; Hoang et al. 2019; Petrovich & Antonini\\n2017). More generally, we can consider the tidal field from\\nboth the SMBH and the cluster which produces a behaviour\\nsimilar to the ZKL mechanism (Petrovich & Antonini 2017;\\nHamilton & Rafikov 2019; Bub & Petrovich 2020; Hamilton\\n& Rafikov 2021).\\nThe strong tidal field due to the SMBH and NSC potentials\\nhas demonstrated to play a key role in the evolution of bina-\\nries in the GC, as it efficiently torques the orbits of these bina-\\nries resulting in near unity eccentricities that lead to mergers.\\nRecent efforts have been made in accounting for different\\ntidal field-driven physical processes in binary evolution such\\nas cluster triaxiality (Bub & Petrovich 2020) and relativistic\\nphase space diffusion (Hamilton & Rafikov 2023), however\\nnone has focused on combining these processes with oth-\\ners such as close encounters with field stars (flybys). Al-\\nthough the dynamics of wide binaries when under the in-\\nfluence of tidal fields (Petrovich & Antonini 2017; Hamil-\\nton & Rafikov 2019a; Rasskazov & Rafikov 2023; Modak &arXiv:2312.17319v1  [astro-ph.GA]  28 Dec 2023', metadata={'source': '/content/bm.pdf', 'page': 0}),\n",
              " Document(page_content='2 M. W INTER -GRANI ´C, C. P ETROVICH , & V. P EÑA-DONAIRE\\nHamilton 2023) and flybys (Collins & Sari 2008; Leigh et al.\\n2017; Michaely & Perets 2020; Hamilton & Modak 2023)\\nhave been studied separately, the potential synergy between\\nboth effects has yet to be investigated in more depth.\\nDue to the high velocity dispersion σin the GC, most bi-\\nnaries in the GC are soft (e.g., σis larger than their Ke-\\nplerian velocity). One single encounter can potentially re-\\nset a binary’s orbit, changing its orientation and even push-\\ning it towards extreme eccentricities. The semi-major axis\\nof these binaries will also tend to increase due to these en-\\ncounters (Heggie 1975; Hills 1975a,b). This orbit widening\\nlimits the efficiency for tidal field driven mergers, and previ-\\nous works have simply prescribed it as a limiting evaporation\\ntimescale tevap(e.g., Stephan et al. 2016; Petrovich & An-\\ntonini 2017; Hoang et al. 2018), using tevapfrom Binney &\\nTremaine (2008). However, beyond the orbit softening, flyby\\ninteractions can produce a random walk in the binary’s an-\\ngular momentum ⃗jwhich may lead to mergers as studied by\\nMichaely & Perets (2020). A study looking into the effect\\nfrom flybys in the cluster-driven dynamics is largely missing\\nin this context (GCs) and the focus of our work.\\nThe presented dynamics is reminiscent of that from wide\\nbinaries in general, including comets in the Oort Cloud . In\\nparticular, (Heisler & Tremaine 1986) studied how the galac-\\ntic tidal field may torque the orbit of binaries and found that\\nflybys can contribute to the diffusion in j, becoming a con-\\nsiderable contribution to the loss rate of comets in the Oort\\nCloud. In contrast, as we show in this paper, binaries in GCs\\nhave slow diffusion rates compared to the torquing rates that\\ndrastically promote the merger fraction.\\nIn this paper we study the binary evolution due to both\\nstellar flybys and tidal fields, proving that by combining both\\nmechanisms can drive these binaries in galactic centers to ex-\\ntreme eccentricities much more efficiently than when these\\nprocesses are considered separately. The paper is organised\\nas follows. We introduce the basic dynamics for binaries in\\ntidal fields in section §2 and discuss the specific case of the\\nGC including flybys in section §3. We present our results\\nfrom numerical simulations in section §4 and we discuss the\\nsynergy found between tidal fields and flybys in §5, along\\nwith the effects on relativistic quenching in section §6. Fi-\\nnally, we present an overall discussion in section §7 and sum-\\nmarize and present the main conclusions in section §8.\\n2.DYNAMICAL EVOLUTION DUE TO TIDAL FIELDS\\nAs binaries are usually immersed in stellar clusters, an im-\\nportant aspect of their dynamical evolution will be driven by\\nthe smooth tidal field of said cluster. These tidal fields drive\\nsecular oscillations in a binary’s eccentricity and inclination,\\nsimilar to the ZKL mechanism. In this section we model\\nthis evolution mathematically, following the work done by\\nHamilton & Rafikov (2019a,b).\\n2.1. Equations of motion\\nWhen studying orbits in axisymmetric potentials, the evo-\\nlution of the inner binary can be approximated by averaging\\nthe tidal tensors over several periods of the outer binary. Thisis known as “torus-averaging”; when the outer orbit of the\\nbinary is not closed, it will trace a non-repeating path around\\nthe cluster which will, after many orbits, densely fill an ax-\\nisymmetric torus. For any function that follows the outer or-\\nbit of the binary, a weighted volume average over this torus\\ncan be used instead of a time average.\\nAs shown by Hamilton & Rafikov (2019a), in this torus-\\naveraged limit the tidal tensor of a given potential Φonly\\nhas diagonal terms, and ⟨Φxx⟩=⟨Φyy⟩. For a smooth po-\\ntential Φthat changes over scales that are much larger than\\nthat of the inner binary’s semi-major axis (expanded up to\\nthe quadrupole tidal field), this results in the torus-averaged\\npotential\\n⟨Φ⟩=3a2\\nin\\n2⟨Φzz+ Φxx⟩\\x141\\n2Γ(5e2\\nz−j2\\nz) +1\\n4e2(1−5Γ)\\x15\\n,\\n(1)\\nwhere\\nΓ≡⟨Φzz−Φxx⟩\\n3⟨Φzz+ Φxx⟩. (2)\\nFrom this potential, a doubly-averaged Hamiltonian can be\\nwritten in terms of ΓasH=HK+H1where HKis the Ke-\\nplerian term and the dimensionless perturbing term is given\\nby (Hamilton & Rafikov 2019b):\\nH1= 2h\\n1−3\\n2e2(5Γ−1)−3Γ(⃗j·ˆjout)2+ 15Γ( ⃗ e·ˆjout)2i\\n,\\n(3)\\nwhere {⃗ e,⃗j}are Milankovitch’s eccentricity and dimension-\\nless angular momentum vectors (e.g., see Tremaine 2023).\\nThe associated dimensionless time is τ=t/τsec, where τsec\\nis the secular timescale given by\\nτ−1\\nsec=3a3/2\\nin\\n2√GM bin⟨Φzz+ Φxx⟩. (4)\\nThe equations of motion for the binary that we use\\nthroughout this work can be written in a vectorial form as:\\nd⃗ e\\ndτ=1\\n2(5Γ−1)(⃗j×⃗ e)−5Γ(⃗ e·ˆjout)(⃗j×ˆjout)\\n+ Γ(⃗j·ˆjout)(⃗ e×ˆjout),(5)\\nd⃗j\\ndτ= Γ(⃗j·ˆjout)(⃗j×ˆjout)−5Γ(⃗ e·ˆjout)(⃗ e×ˆjout),(6)\\nwhere we shall assume that ˆjoutremains fixed throughout the\\nevolution. The latter approximation assumes that either the\\ntidal field is spherically symmetric or the outer orbit lies in\\nthe meridional plane of an axisymmetric field (Petrovich &\\nAntonini 2017).\\nAs discussed by Hamilton & Rafikov (2019b,a), Γde-\\ntermines the phase-space structure of the Hamiltonian, and\\nhence is a useful parameter to classify different dynamical\\nregimes for binary evolution. For example, Γ = 1 when the', metadata={'source': '/content/bm.pdf', 'page': 1}),\n",
              " Document(page_content='BINARY EVOLUTION PERTURBED BY FLYBYS AND TIDAL FIELDS 3\\npotential is Keplerian (only a SMBH at the center), and hence\\nthe Hamiltonian in equation (3) reduces to the quadrupole\\nZKL Hamiltonian. The secular timescale reduces to the well-\\nknown ZKL timescale (e.g., Holman et al. 1997):\\nτZKL=4a3\\nout(1−e2\\nout)3/2M1/2\\nbin\\n3a3/2\\ninMBHG1/2\\n≃3×105\\x12Mbin\\n10M⊙\\x13\\x12MBH\\n4×106M⊙\\x13−1\\n(7)\\n×\\x10ain\\n25au\\x11−3/2\\x12aout(1−e2\\nout)1/2\\n0.3pc\\x133\\nyear.\\n(8)\\nSimilarly, we can obtain the Hamiltonian for a binary evolv-\\ning in a thin Galactic disk by replacing Γ = 1 /3(see discus-\\nsion in §7.1), while realistic spherical potentials can be re-\\nproduced with 0⩽Γ⩽1as shown by Hamilton & Rafikov\\n(2019a).\\nAn important property of the secular Hamiltonian in Equa-\\ntion (3) is its axissymmetry (i.e., invariability to rotations\\naround ˆjout), which implies the z-component of the angu-\\nlar momentum ∝p\\nain(1−e2\\nin) cosiwithcosiin=ˆjin·ˆjout\\nis conserved. Thus, the Hamiltonian is integrable and we\\ncan express the maximum eccentricity due to tidal fields as\\n(Hamilton & Rafikov 2019a):\\nemax=r\\n1−10Γ cos2i0\\n5Γ + 1, (9)\\nwhere i0is the initial initial when e2\\n0is nearly 0.\\nFinally, in addition to the equations of motion (5) and (6),\\nwe add an extra term in Equation (5) to account for the rela-\\ntivistic precession given by\\nd⃗ e\\ndt\\x0c\\x0c\\x0c\\nGR=˙ωGR\\n(1−e2)3/2⃗j×⃗ e, (10)\\nwhere\\n˙ωGR=3G3/2M3/2\\nbin\\na5/2\\ninc2. (11)\\n2.2. Merger conditions\\nA major application we consider in the galactic center is\\nthe evolution and potential merger of compact object bina-\\nries. When the two components of these binaries come very\\nclose together, they lose energy by GW radiation. The merger\\ntimescale for this gravitational radiation is defined by Peters\\n(1964) as\\nτGW=3\\n85\\x12a4\\ninc5\\nG3m1m2Mbin\\x13\\n(1−ein)7/2. (12)\\nIf a binary’s orbit shrinks significantly in one secular cycle,\\nthen a rapid merger will take place. For this to happen, the\\ntimescale for the change in the semi-major axis of the binarydue to GW losses ( τGW) must be shorter than the timescale\\nfor the change in the pericentric distance ain(1−ein)due to\\ntidal fields, which, at high eccentricities, is ≃√\\n1−e2τsec.\\nThis results in the condition\\n1−emerger≲3×10−5\\x12aout(1−e2\\nout)1/2\\n0.1pc\\x13\\n, (13)\\nfor binaries with Mbin= 10M⊙initially at ain= 10 au when\\nonly ZKL cycles are considered (only SMBH). A similar ex-\\npression can be obtained for a more general cluster potential\\n(arbitrary Γ; see Equation 16 for a secular timescale adding\\na cluster with a Hernquist potential).\\nIn order for BH (stellar) binaries to reach extremely high\\neccentricities of 1−emax≲10−5(1−emax≲10−3), we\\nrequire that ˙ωGRτsec≲0.01(˙ωGRτsec≲0.1) as shown by\\nPetrovich & Antonini (2017). In general, we will define ex-\\ntreme eccentricities as 1−emax≲10−4following equation\\n(13), hence binaries that achieve these values will be consid-\\nered as possible mergers.\\n3.DYNAMICS IN THE GALACTIC CENTER\\nIn the case of a spherical γ-family cluster with a distance\\nscale sand with a central SMBH in which we have a circular\\norbit, Γbecomes\\nΓ =3MBH+Mcla3−γ\\nout(aout+s)γ−4(3aout+sγ)\\n3[MBH+Mcla3−γ\\nout(aout+s)γ−4(aout+s(4−γ))],\\n(14)\\nwhere we have considered the tidal averaged tidal tensors\\n⟨Φxx⟩and⟨Φzz⟩as derived by Bub & Petrovich (2020).\\nIn the simplified case that we consider a spherical Hern-\\nquist potential with γ= 1to model the center of our galaxy,\\nthis reduces to\\nΓ =3MBH+Mcla2\\nout(aout+s)−3(3aout+s)\\n3[MBH+Mcla2out(aout+s)−3(aout+ 3s)], (15)\\nwhere sis the scale radius of the potential.\\nFor small values of aout/swe have Γ≈1, where the\\nSMBH dominates over the cluster and hence the dynamical\\nevolution of the binary reduces to the ZKL mechanism as\\nmentioned in section §2.1. The same happens when aout/s\\nbecomes very large; the cluster and SMBH in this limit be-\\nhave as a single body so that as we get further away from the\\ncentral SMBH, Γ→1.\\nThe secular timescale in these kinds of environments then\\nbecomes (Bub & Petrovich 2020):\\nτ−1\\nsec=3a3/2\\nin\\n2√GM bin \\nGM BH\\n2a3out\\n+GM cl\\naγ\\nout(aout+s)3−γ\\x14\\n2−3aout+sγ\\n2(aout+s)\\x15!\\n.(16)', metadata={'source': '/content/bm.pdf', 'page': 2}),\n",
              " Document(page_content='4 M. W INTER -GRANI ´C, C. P ETROVICH , & V. P EÑA-DONAIRE\\nFigure 1. Full evolution of a binary due to both tidal fields and flyby effects. The initial parameters are aout= 0.3pc,ain= 25 AU,\\nein= 0.1,ωin=π/4,Ωin= 0,iin= 80 °. Each stellar component of the binary has a mass of 5M⊙, and we consider the fiducial scenario of\\nMBH= 4×106M⊙. The red curves represent the evolution in a flyby free scenario, by simply integrating Equations (5) and (6). In panels (d)\\nand (e), the shaded region contains the median and both upper and lower quartiles.\\n3.1. Fly-bys\\nThe main focus of this work is to study the additional effect\\nproduced by flybys on the evolution of binaries in tidal fields.\\nDue to the high velocity dispersion in the GC, the binaries\\nresiding there are mostly soft ( vorb< σ, with σthe velocity\\ndispersion of the cluster), hence we expect flyby encounters\\nto produce a systematic drift in the semi-major axis of the\\nbinary. This is a consequence of the Heggie-Hills law, de-\\nrived using thermodynamic arguments: \"hard binaries tend to\\nharden, whereas soft binaries tend to soften\" (Hills 1975a,b;\\nHeggie 1975).\\nIn order to classify binaries between hard and soft, we can\\nwrite the velocity dispersion in the galactic center as accord-\\ning to Kocsis & Tremaine (2011)\\nσ(r) =\\uf8f1\\n\\uf8f4\\uf8f4\\uf8f2\\n\\uf8f4\\uf8f4\\uf8f3280kms−1p\\n0.1pc/rp\\n1−0.035(r/0.1pc)2.2,\\nifr <0.22pc\\n250kms−1p\\n0.1pc/r, ifr >0.22pc\\n(17)\\nThe time between flyby encounters\\ntenc= (n⋆σpvp)−1, (18)\\nwhere σp=πb2\\nmaxis the geometric cross section considered\\nfor the flybys and n⋆is the stellar number density of the en-\\nvironment. For the galactic center it is considered as n⋆≈106pc−3(Hamers & Tremaine 2017) and n⋆≈0.1pc−3for\\nthe solar neighbourhood. It can be more accuratelly modelled\\nby (Kocsis & Tremaine 2011)\\nn⋆(r) =2.8×106M⊙pc−3\\n⟨m⋆⟩\\x12r\\n0.22pc\\x13−γ\\n, (19)\\nwithγ= 1.2inside 0.22pc and γ= 1.75outside 0.22pc.\\nWe take ⟨m⋆⟩ ≈ 1M⊙as the average stellar mass in the\\nGC; Alexander & Pfuhl (2013) presents equations that result\\nin⟨m⋆⟩ranging from 1M⊙to2M⊙, therefore to simplify\\ncalculations we take all flybys to have a solar mass.\\n4.NUMERICAL SIMULATIONS\\nBecause the interaction time due to a flyby (order ain/σ,\\n∼0.2years) is much shorter than the secular timescale, we\\nevolve an individual binary using the secular equation of mo-\\ntions in (5) and (6) subject to discrete instances where a flyby\\ntakes place, modifying the orbital elements of the binary, and\\ncontinue the secular evolution until the next flyby takes place.\\nIn other words, we switch back and forth between tidal fields\\nand flybys.\\nThe flybys are modelled using direct N-body integrations\\nwith REBOUND (Rein & Liu 2012), with an encounter rate\\nas specified in equation (18) defining bmax= 5ainto avoid\\nconsidering flybys that are too distant. This definition allows', metadata={'source': '/content/bm.pdf', 'page': 3}),\n",
              " Document(page_content='BINARY EVOLUTION PERTURBED BY FLYBYS AND TIDAL FIELDS 5\\nus to stay close to the regime in which the impulse approx-\\nimation is valid, which will prove to be relevant in section\\n§5. We repeated our fiducial numerical experiments in which\\nwe doubled bmaxto10ain, and found that the outcome was\\nstatically similar to those with bmax= 5ain, plus the fact that\\na higher value for bmaximplies a higher frequency for flyby\\nencounters, which is much more computationally expensive\\nto simulate.\\nWe establish the velocity of our flybys to be constant at\\nvp= 200 kms−1, considering that within 1 pc of the center\\nof our galaxy σvaries roughly between 100 and 300 kms−1\\nas according to equation (17). These simplifications allow for\\ncontrolled simulations, which makes comparisons with ana-\\nlytical estimates such as the ones made in section §5 easier.\\nFigure 1 shows the complete dynamical evolution of an\\nindividual binary with aout= 0.3pc. The red curves show\\nthe evolution of an identical binary in a flyby-free scenario,\\nwhere we only get tidal field driven evolution. It is quite clear\\nby comparing these curves with our results that the inclusion\\nof flybys in our model results in a more dramatic evolution\\nwhich involves extreme eccentricities, orbital flips and jumps\\nbetween the ein−ωinphase-space curves.\\nThe individual changes in ainpresented in panel (a) follow\\na random walk, but show an overall tendency of making the\\nbinary wider. This is expected due to the Heggie-Hills law,\\nas mentioned earlier. We can also easily see in panel (b) that\\nby only taking into account tidal field driven evolution, such\\nhigh eccentricities would never have been achieved by this\\nparticular binary. However, this becomes possible when in-\\ncluding flybys in the model, which allows the binary to reach\\nvalues of 1−ein≈10−5. As discussed in section §2.2, with\\nthese extreme eccentricities a rapid merger can be produced\\ndue to GW tidal captures.\\nIt is also interesting to note that in panel (c) we can see or-\\nbital flips, in which the orbit switches between prograde and\\nretrograde. Considering that the Hamiltonian shown in Equa-\\ntion (3) is only of quadrupole order and assumes eout= 0,\\nthese flips should not be possible as jz=p\\n1−e2\\nincosiinis\\nfixed (e.g., Naoz 2016). However, as flybys modify both ein\\nandiindriving a slow random walk in jz(see also the exam-\\nples in Figure 5), the orbit is allowed to flip, a phenomenon\\nobserved during the high eccentricity episodes.\\nPanels (d) and (e) show the variation in ωinandiinproduct\\nof each flyby encounter. We can see that both these varia-\\ntions in general are very small, however, along with the vari-\\nations in eccentricity due to each encounter, they are enough\\nto make the binary’s orbit jump from one phase-space curve\\nto another as is shown in panel (f). Since flybys not only\\naffect the magnitude of the eccentricity but also the orienta-\\ntion of the orbit, we get changes both in ωinand in einwhich\\nproduce these ‘jumps’ in the phase-space portrait. In particu-\\nlar, in a flyby-free scenario the orbit would librate around the\\nfixed point ωin=π/2, however by including flybys in our\\nmodel the orbit generally circulates one as shown in panel\\n(f).\\n4.1. Population statistics\\nFigure 2. CDFs of the maximum eccentricities reached in the galac-\\ntic center after 10, 25, 50 and 100 initial secular timescales τ0, as\\nlabeled. The dashed histogram represents the CDF for a flyby free\\npopulation, evolved for 100τ0.\\nIn order to quantify the overall effects of tidal fields and\\nflybys on binary evolution, we generated a series of simpli-\\nfied population synthesis models. It is important to note that\\nall binaries within the population are independent from one\\nanother, hence we are not considering possible effects due to\\nbinary-binary interactions. We also consider a dynamical sta-\\nbility criterion, so as to not consider binaries that become ex-\\ntremely wide and hence unstable when (Grishin et al. 2016)\\nain>0.25aout\\x12Mbin\\nMBH\\x131/3\\n. (20)\\nIn Figure 2 we show the CDF of the maximum eccentrici-\\nties achieved by a population of ∼700binaries in the galactic\\ncenter at different stages of their evolution. The initial eccen-\\ntricities of these binaries follow a thermal distribution such\\nthatf(e)de= 2ede, and their initial inclinations are uniform\\nsuch that cosi0is also uniform between -1 and 1. The ec-\\ncentricity of the outer orbits is set to 0. All binaries are twins\\nwith a total mass of 10M⊙and are evolved for 100τ0,τ0\\nbeing the initial secular timescale of each individual binary\\naccording to equation (16). We consider the fiducial scenario\\nof our GC, where MBH= 4×106M⊙,Mcl≈8MBH, and\\ns= 4pc.\\nAfter only a few secular timescales, the initial thermal ec-\\ncentricity distribution is modified into a wider distribution,\\nachieving higher eccentricities as expected from the tidal\\nfields. We can also observe the cumulative nature of this type\\nof dynamical evolution; the longer we let the binaries evolve,\\nthe higher the probability of them reaching extreme eccen-\\ntricities. This is expected from a chaotic nature of the binary\\nevolution depicted in Figure 1.\\nIn Figure 3 we can see the CDFs of the maximum eccen-\\ntricities achieved after 100τ0by three different populations\\nof binaries. One population was evolved considering only', metadata={'source': '/content/bm.pdf', 'page': 4}),\n",
              " Document(page_content='6 M. W INTER -GRANI ´C, C. P ETROVICH , & V. P EÑA-DONAIRE\\nFigure 3. CDFs of the maximum eccentricities and minimum peri-\\ncenter distances qin=ain(1−ein)reached in the galactic center\\nafter 100 initial secular timescales τ0for different dynamical evolu-\\ntion mechanisms: tidal fields, flybys and a combination of both.\\ntidal field effects, the second considering only flybys and\\nthe third taking into account both effects (the same popula-\\ntion as shown in figure 2). The figure shows that tidal fields\\nare the least efficient dynamical mechanism in producing ex-\\ntreme eccentricities; only about ∼1%of binaries evolved\\nwith tidal fields alone reach 1−emax<10−4. Flybys on the\\nother hand prove to be slightly more effective, driving about\\n3%of binaries in the population to said values. However,\\nwhen combining both mechanisms a clear synergy is found,\\nincreasing this probability up to ∼34%. For reference,\\nthe bottom panel shows the minimum pericenter distances\\nfor the same population of binaries. Although several bina-\\nries reach very high eccentricities, as was mentioned in sec-\\ntion §3.1 these binaries will tend to get softer, which means\\nthatainand hence the pericentric distance q=ain(1−ein)\\nwill grow as well. This is where the stability criterion men-\\ntioned previously in equation (20) becomes important. As\\ncan be seen in the CDF, the extreme values of the eccen-\\nFigure 4. CDFs of the maximum eccentricities achieved by diffu-\\nsion in different orbital parameters, evolved for 50τ0. The black\\ndashed curve represents evolution considering diffusion in all the\\norbital parameters due to flybys; it is the same as the yellow curve\\nshown in figure 2.\\ntricities achieved seem to dominate over the increase in ain,\\nproducing a considerable amount of binaries with pericen-\\ntric distances smaller than ∼10−3au, equivalent to a merger\\ntimescale τGW<3×105yrs (Eq. [12]). The latter may be\\nshorter than the ZKL timescale in Equation (7) and, there-\\nfore, expected to lead to a merger within one secular cycle.\\n5.TIDES VS. FLYBYS\\nIn this section we analyze in more depth the diffusive ef-\\nfects of flybys on the binary evolution.\\nAs discussed in the previous section, the flybys impart ran-\\ndom changes in all orbital elements ( e−i−ω), but it remains\\nunclear whether the diffusion in emaxis dominated by the be-\\nhavior of a single element or it is a collective process. As a\\ncontrolled experiment, we show in Figure 4 the CDFs of the\\nmaximum eccentricities achieved by populations in which\\nflyby encounters were replaced by random changes in indi-\\nvidual orbital elements using the statistical distribution from\\nN-body experiments. We clearly observe that diffusion in the\\ninclination is the main effect driving binaries to high eccen-\\ntricities. This translates into diffusion of jzas the dominant\\ndriver of maximum eccentricity grows. In turn, the diffusion\\nofωinandeinadds little compared to the case where no dif-\\nfusion occurs (i.e., tidal fields only).\\nGuided by these numerical experiments, we define a pa-\\nrameter Dthat quantifies the average diffusion rate of the\\nspecific angular momentum vector due to a single flyby en-\\ncounter as\\nD=⟨∥∆⃗j∥2⟩(f,b,i,ω, Ω)\\ntenc. (21)\\nThis coefficient not only includes changes in the specific an-\\ngular momentum j=√\\n1−e2, but also changes in its ori-\\nentation which is relevant to model the changes in jz. As', metadata={'source': '/content/bm.pdf', 'page': 5}),\n",
              " Document(page_content='BINARY EVOLUTION PERTURBED BY FLYBYS AND TIDAL FIELDS 7\\nFigure 5. Evolution of orbital parameters for three binaries with different values of R, all with the same initial conditions of ain= 25 au,\\naout= 0.3pc,ein=0.1 and eout= 0. All were evolved for 50τ0,τ0being the initial secular timescale for each binary. Note the difference in the\\nscale of the vertical axis in the top row; as Rincreases, aingrows more drastically.\\nexplained in Appendix A.1, the ⟨·⟩average is carried over\\nan ensemble of binaries with random orientations and phases\\nrelative to the stellar perturber.\\nFrom this diffusion coefficient we define a dimensionless\\nparameter Rthat quantifies the expected diffusion in ⃗jdue\\nto flybys after a secular timescale τ0has occurred as\\nR=p\\nDτ0. (22)\\nThus, the limits R → 0andR → ∞ refer to scenarios com-\\npletely dominated by tidal fields and by flybys, respectively.\\nIn Appendix A.1 we show the full derivation of D, in which\\nwe have used the impulse approximation to find an analyticalexpression for ∆⃗j. From this we obtain\\nR ≃0.02\\x12Mbin\\n10M⊙\\x13−1\\n4\\x12Mp\\nM⊙\\x13\\x12MBH\\n4×106M⊙\\x13−1\\n2\\n×\\x10vp\\n200km/s\\x11−1\\n2\\x10ain\\n25au\\x113\\n4\\x12aout\\n0.3pc\\x133\\n2\\n×\\x12n⋆\\n106pc−3\\x131\\n2\\x12f(β)\\n1.8\\x131\\n2\\n,(23)\\nwhere β=s/a outandf(β)is defined in equation (A8).\\nThe accuracy of this expression was verified numerically,\\nby simulating the diffusion in a binary’s angular momen-\\ntum vector due to only flyby interactions using REBOUND.\\nFigure A2 shows that the numerical equivalent for Ralso\\nconverges to 0.02when considering the fiducial scenario of\\nMbin= 10M⊙,Mp=M⊙,MBH= 4×106M⊙,ain= 25 au\\nandaout= 0.3pc.', metadata={'source': '/content/bm.pdf', 'page': 6}),\n",
              " Document(page_content='8 M. W INTER -GRANI ´C, C. P ETROVICH , & V. P EÑA-DONAIRE\\nFigure 6. Merger fraction as a function of the diffusion parameter\\nR. The dashed lines indicate the merger fractions for the limits\\nR= 0 (tidal fields alone) and R=∞(flybys alone). The error\\nbars are evaluated assuming a Poisson distribution as√\\nN,Nbeing\\nthe amount of mergers for each bin.\\nIn general, we expect different types of evolution when\\nvarying R, as it should become much more diffusive and in-\\ncoherent as this parameter grows. Figure 5 shows this be-\\nhavior, where we have evolved three binaries with different\\nvalues of Rby varying MBHin each case. All three binaries\\nhave initial parameters of ain= 25 au,aout= 0.3pc,ein= 0.1\\nandeout= 0, and were evolved for 50τ0where τ0is the\\ninitial secular timescale for each case according to equation\\n(16). It is important to note that Rincreases with τsec, there-\\nfore in a more diffusive scenario we will have more flybys\\nper secular cycle as the cycles grow longer. We can see in\\nthe rightmost column how coherent cycles due to tidal fields\\nbecome much more difficult to identify when R= 0.06, as\\nopposed to the case where flybys start to become less impor-\\ntant as shown in the first column where R= 0.006. Panel\\n(d) more specifically shows a much less chaotic evolution in\\nwhich we can easily distinguish coherent cycles with some\\nperturbations such as extreme eccentricities and variations in\\nthe secular timescale, as opposed to panel (f) where the cy-\\ncles are greatly perturbed by the flybys and hence such large\\neccentricities are not achieved.\\nThe bottom row shows the z−component of the angular\\nmomentum jzwhich should be a conserved quantity in a\\nflyby-free scenario, however we can see how its value drifts\\ndue to the diffusion caused by flybys, especially as Rin-\\ncreases. Due to the increasingly chaotic nature of the evolu-\\ntion of binaries in more diffusive regimes, it becomes much\\nharder to reach the extreme eccentricities required to produce\\na merger, as can be seen in the case for R= 0.06. The\\nlarge amount of flybys per secular timescale constantly re-\\nsets the orbital parameters of the binary, preventing it from\\ncompleting regular eccentricity cycles and hence decreasing\\nthe chances of achieving eccentricities of 1−e≲10−4.\\nFigure 7. Evolution of a binary that begins in a GR quenched\\nregime in which mergers are forbidden (colored region). Flyby in-\\nteractions allow this binary to become softer and move out of this\\nregime, hence becoming a merger candidate. The top panel shows\\nthe evolution of ϵGRas according to equation (10), and the bottom\\npanel shows the evolution of ainandqin.\\n5.1. Experiments changing R\\nIn Figure 6 we show the merger fractions of different bi-\\nnary populations as a function of R, which was varied by\\nchanging the value of MBH. These populations were all in-\\ntegrated for ∼15Myr, which is roughly 50τsecfor a binary\\nstarting with ain= 25 au at 0.3pc considering the fiducial\\ncase of R= 0.02. We consider any binary that reaches\\n1−emax≤10−4to be a merger, as was discussed in sec-\\ntion §2.2. As mentioned previously, a less diffusive regime\\ncomes with shorter secular timescales, therefore by integrat-\\ning a population with R= 0.006for15Myr we are actually\\nletting them evolve for ∼500τsec. However as we keep low-\\neringRby increasing MBH, binaries start to become unstable\\nvery fast as according to equation (20) and hence the likeli-\\nhood of them merging before they evaporate decreases. We\\nalso move into a regime in which the binaries only experi-\\nence a few flyby interactions in each cycle, which could also', metadata={'source': '/content/bm.pdf', 'page': 7}),\n",
              " Document(page_content='BINARY EVOLUTION PERTURBED BY FLYBYS AND TIDAL FIELDS 9\\nFigure 8. Contours of the diffusion parameter Rin Equation (23) as a function of ainandaout, considering the fiducial scenario of MBH=\\n4×106M⊙,Mp= 1M⊙andMbin= 10M⊙. The boundary for unstable binaries is plotted as according to Equation (20), while the limit for\\nhard binaries is defined by vkep> σ. We also plot the region in which relativistic quenching ( ϵGR>0.1) does not allow for mergers.\\ncontribute in decreasing the probability of reaching extreme\\neccentricity values as occurs in figure 6. Our experiments\\nsuggest that there is an optimal value for R ∼ 0.02at which\\nthe merger fraction peaks.\\n6.LIFTING THE RELATIVISTIC QUENCHING FROM\\nFLYBYS\\nConsidering the properties of the GC, it is likely that a con-\\nsiderable amount of binaries residing in it will be subject to\\nrelativistic quenching. This presents an obstacle when pro-\\nducing mergers, as it does not allow binaries to reach ex-\\ntreme eccentricities. We can quantify the relative importance\\nof relativistic precession using the following dimensionless\\nparameter ϵGRas (Petrovich & Antonini 2017):\\nϵGR≡τsec\\nτGR\\n=4GM2\\nbina3\\nout(1−e2\\nout)3/2\\nc2a4\\ninMBH\\n×\"\\n1 +Mcl\\nMBH2a3−γ\\nout\\n(aout+s)3−γ\\x12\\n2−3aout+sγ\\n2(aout+s)\\x13#−1\\n,\\n(24)\\nwhere binaries with values of ϵGR>0.1are considered as\\nquenched.\\nWhen this parameter becomes smaller relativistic quench-\\ning can be lifted, allowing for a potential merger driven by\\nthe tidal fields. This can be achieved through the softening\\nof binaries, as this will increase their secular timescale τsec.\\nConveniently, this paper has focused on the effect of flybys\\non binaries in the GC, which tend to soften said binaries and\\nhence can effectively lift the effects of relativistic quenching.In figure 7 we can see a specific example where soften-\\ning driven by flybys lifts relativistic quenching and allows\\nfor a merger. The initial conditions of the simulated binary\\nareain= 1au and aout= 0.3pc, which makes it a soft bi-\\nnary severely quenched by GR with an initial dimensionless\\nparameter ϵGR∼200. However, due to flyby interactions\\nincreasing ain,ϵGRreaches values smaller than 0.1 hence lift-\\ning relativistic quenching. It is clear that while subject to\\nGR quenching the eccentricity of the binary’s orbit remains\\npractically constant at a low value, such that ain∼qin. How-\\never once it escapes the GR quenching regime, extreme ec-\\ncentricity oscillations are induced such that the pericentric\\ndistance reaches values as low as 10−3au, allowing for a po-\\ntential merger.\\n7.DISCUSSION\\nWe have shown that stellar flybys can greatly enhance the\\nmerger rates of binaries in the centers of galaxies by assisting\\nthe eccentricity growth to extreme values that are driven by\\nthe tidal torques exerted by the SMBH and the central cluster.\\nMore specifically, our results demonstrate that the diffu-\\nsive effects of flybys and the torquing of binary’s orbits due\\nto tidal fields produce comparable merger rates (within a fac-\\ntor of 3; see Figure 3). In turn, for regimes in which the\\ndiffusion rate is much slower than the tidal field torquing rate\\n(R ∼ 0.01; see §5) flybys added to the tidal fields are able\\nto produce a merger rate ∼30times larger that the one pro-\\nduced by tidal fields alone. This is due to the chaotic behav-\\nior introduced by flyby interactions, as they constantly reset\\nthe binary’s orbital parameters and consequently can produce\\nrandom and dramatic changes in the orbit. Most importantly,\\nthe diffusion of the z−component of the angular momentum\\njz.', metadata={'source': '/content/bm.pdf', 'page': 8}),\n",
              " Document(page_content='10 M. W INTER -GRANI ´C, C. P ETROVICH , & V. P EÑA-DONAIRE\\nWe also find that the softening of GR quenched binaries\\ndue to flyby interactions allows them to escape the relativistic\\nquenching regime, thus expanding the available phase-space\\nfor mergers.\\n7.1. Dynamical regimes in the galactic center\\nIn Figure 8 we show the different dynamical regimes as a\\nfunction of ainandaout, along with a contour plot of R. It\\nis important to note that in this plot, Equation (23) has been\\nadapted such that vp,n⋆andf(β)are written in terms of\\naout. We can see that for binaries that are extremely tight\\n(ain≲0.1au),Rreaches very low values which implies an\\nalmost insignificant diffusion in the angular momentum. If\\nthese binaries are subject to relativistic quenching, even if\\nthey are soft, it is unlikely that flyby interactions will have\\nenough impact on their orbit to push them out of this regime.\\nConsidering this, the lifting of GR quenching due to flyby\\ninteractions is probably most efficient when ain≳0.1au\\nOur results suggest that previous studies such as Petro-\\nvich & Antonini (2017) and Hoang et al. (2018) (who con-\\nsidered a regime similar to our own with aout≲0.5pc and\\nϵGR≲0.1) have underestimated the effect of interactions\\nwith field stars, presenting it as merely a limiting evapora-\\ntion timescale that should inhibit merger rates. Our findings\\non the other hand, suggest that flyby interactions are actually\\nquite efficient when producing mergers, much more so than\\ntidal fields alone. This is due mainly to the diffusion in the\\nangular momentum vector, quantified by our dimensionless\\nparameter R, that for these previous works would be of order\\nR ∼ 0.01. This, combined with the ability to lift relativistic\\nquenching and hence increase the phase-space for mergers\\n(which was not taken into account in the mentioned studies\\neither), provides strong evidence for the importance of flybys\\nin producing mergers in the GC and shows that their effects\\nmust be accounted for.\\n7.2. Other astrophysical environments\\nOur results are applicable the evolution of wide binaries\\nin various astrophysical environments. Next we discuss our\\nresults in the context of previous results.\\nSolar neighborhood —The dynamical evolution of wide bina-\\nries is equivalent to that of comets in the Oort Cloud studied\\nby Heisler & Tremaine (1986). This is a different regime than\\nthe one considered in this paper, where the evolution should\\nbe flyby dominated (higher value for R) due to the orbits of\\nbinaries being wider and having longer secular timescales.\\nWe can write an expression for Rin this environment as fol-\\nlows\\nR ≃9\\x12Mbin\\nM⊙\\x13−1\\n4\\x12Mp\\n0.162M⊙\\x13\\x10vp\\n40km/s\\x11−1\\n2\\n×\\x12ain\\n2.5×104au\\x133\\n4\\x12n⋆\\n1.14pc−3\\x131\\n2\\x12ρ0\\n0.185M⊙pc−3\\x13−1\\n2\\n,\\n(25)\\nwhere we have considered the secular timescale asτg=M1/2\\nbin\\n2πρ0G1/2a3/2\\nin, (26)\\nwithρ0= 0.185±0.02M⊙pc−3being the local density as\\nused by Heisler & Tremaine (1986).\\nWe can see that for an average comet in the Oort Cloud we\\ngetR ≃ 9as opposed to the R ≃ 0.02obtained for compact\\nobject binaries in the GC. This implies that in the Oort Cloud\\nflybys dominate over tidal fields, as the diffusion rate from\\nthese interactions is ∼9times faster than the torquing due\\nto the tidal field. This is mainly due to the difference in the\\namount of flybys in a secular timescale, τ0/tenc. As shown\\nin equation (A7), we get about 68 encounters for an average\\nbinary in the GC, while in the solar neighbourhood the aver-\\nage comet suffers ∼2×104flyby encounters (considering\\nequation (26) as τ0andtenc≃0.015Myr as used by Heisler\\n& Tremaine 1986). This increase in the amount of close en-\\ncounters inevitably results in more diffusion of the angular\\nmomentum ⃗j, hence the higher value of Robtained in this\\nregime. Consistently, the numerical experiments by Heisler\\n& Tremaine (1986) show the secular evolution is largely de-\\ntuned due to the stellar flybys, making it hard to complete a\\nclean secular cycle in the ω−espace (see figure 4 therein).\\nGlobular clusters —A recent study by Rasskazov & Rafikov\\n(2023) looked at the evolution of compact object binaries\\nwhen subject to tidal fields and stellar encounters in glob-\\nular clusters, similar to the work done throughout this paper.\\nThe main difference is that these globular clusters are less\\ndense and do not contain a central SMBH as the NSCs we\\nhave considered in this study, which means that they have a\\nmuch lower velocity dispersion. This results in a higher ain\\nhard-soft limit for binaries, implying that many of the bina-\\nries in these environments are actually hard, unlike our work.\\nThis is particularly important when speaking of GR quench-\\ning, as we have found that flybys are able to diminish this\\neffect only when they are capable of softening binaries. In\\nthe case of hard binaries, flybys should tend to promote GR\\nquenching rather than lift it, further inhibiting merger rates.\\nBy looking at the behavior in the various evolution exam-\\nples in Rasskazov & Rafikov (2023), we suggest that sim-\\nilar to the galactic field, the diffusion parameter Ris large\\n(dominated by flybys). We note that equation (23) is not ap-\\nplicable here because the encounters are generally far from\\nbeing impulsive given the low dispersion velocities. More\\nwork would need to be done to extend the analysis to these\\nenvironment, including flybys that can be secular in nature\\n(e.g., Heggie & Rasio 1996; Hamers & Samsing 2019).\\n8.CONCLUSIONS\\nIn this work we have studied the combined effect of cluster\\ntidal fields and flyby interactions on the evolution of binaries\\nin the galactic center. Our main result is the dramatic synergy\\nbetween these two physical processes at driving binaries into\\nextremely high eccentricities ( 1−emax≲10−4) and their\\nsubsequent merger.', metadata={'source': '/content/bm.pdf', 'page': 9}),\n",
              " Document(page_content='BINARY EVOLUTION PERTURBED BY FLYBYS AND TIDAL FIELDS 11\\nAlthough stellar flybys tend to increase the semi-major\\naxis of the binaries thus inhibiting the mergers, they also give\\nrise an stochastic evolution of other orbital elements that can\\ndramatically shrink the pericenter distance before the binary\\nbecomes unbound. This dynamics has been previously stud-\\nied showing that it could lead to mergers of wide binaries in\\nthe galactic field, consistent with our results for the galactic\\ncenter including only flybys that drive mergers in few a per-\\ncent of the systems. We extend these results and show that\\nthis rate increases by an order of magnitude when flybys act\\nin concert with the static tidal field from the SMBH and the\\ncentral cluster.\\nWe find that the main cause of this synergy is the persistent\\ntidal field-driven eccentricity excitation that is enhanced by\\nthe diffusion of jzdue to flyby encounters. We calculated the\\ndiffusion rate of the angular momentum vector ⃗jdue to fly-\\nbys using the impulse approximation and assess the various\\nregimes with a dimensionless parameter R(Eq. 23) that mea-\\nsures the expected diffusion within a secular timesale (i.e,\\ntorquing timescale due to tidal fields). Our experiments sug-\\ngest that the merger rates peak in the range R ∼ 0.01−0.1,\\nmeaning when the diffusion rate is ∼10−100times more\\nslow than the torquing due to the tidal field. This regime is\\ntypical of the central parsec of our GC (see Figure 8), while\\nother environments such the solar neighborhood and stellar\\nclusters tend to have much higher values of R.We also show that the gradual softening of binaries in the\\nGC can lift the relativistic quenching of initially tight bina-\\nries, thus expanding the available phase-space for mergers.\\nThis is also likely to contribute to the previously discussed\\nsynergy, given that by increasing the amount of binaries that\\ncan potentially merge in a population we can expect a higher\\nmerger rate.\\nIn summary, we conclude that despite the gradual soften-\\ning of binaries due to stellar encounters, these greatly en-\\nhance merger rates in GCs by promoting the tidal field driven\\neccentricity excitation This behavior has been ignored in pre-\\nvious works and further reinforces that galactic centers are\\nideal environments for the production of compact object bi-\\nnary mergers.\\nACKNOWLEDGMENTS\\nWe would like to thank Antranik Sefilian, Carolina Char-\\nalambous, and Diego Muñoz for useful discussions. CP ac-\\nknowledges support from CATA-Basal AFB-170002, ANID\\nBASAL project FB210003, FONDECYT Regular grant\\n1210425, CASSACA grant CCJRF2105, and ANID+REC\\nConvocatoria Nacional subvencion a la instalacion en la\\nAcademia convocatoria 2020 PAI77200076.\\nREFERENCES\\nAbbott, B. P., Abbott, R., Abbott, T. D., et al. 2016, PhRvL, 116,\\n061102, doi: 10.1103/PhysRevLett.116.061102\\n—. 2017, PhRvL, 119, 161101,\\ndoi: 10.1103/PhysRevLett.119.161101\\nAlexander, T., & Pfuhl, O. 2013, The Astrophysical Journal, 780,\\n148, doi: 10.1088/0004-637x/780/2/148\\nAntonini, F., & Perets, H. B. 2012, The Astrophysical Journal, 757,\\n27, doi: 10.1088/0004-637x/757/1/27\\nBinney, J., & Tremaine, S. 2008, Galactic Dynamics: Second\\nEdition\\nBub, M. W., & Petrovich, C. 2020, ApJ, 894, 15,\\ndoi: 10.3847/1538-4357/ab8461\\nCollins, B. F., & Sari, R. 2008, The Astronomical Journal, 136,\\n2552, doi: 10.1088/0004-6256/136/6/2552\\nGeorgiev, I. Y ., & Böker, T. 2014, Monthly Notices of the Royal\\nAstronomical Society, 441, 3570, doi: 10.1093/mnras/stu797\\nGhez, A. M., Salim, S., Hornstein, S. D., et al. 2005, The\\nAstrophysical Journal, 620, 744–757, doi: 10.1086/427175\\nGillessen, S., Eisenhauer, F., Trippe, S., et al. 2009, ApJ, 692,\\n1075, doi: 10.1088/0004-637X/692/2/1075\\nGrishin, E., Perets, H. B., Zenati, Y ., & Michaely, E. 2016,\\nMonthly Notices of the Royal Astronomical Society, 466, 276,\\ndoi: 10.1093/mnras/stw3096Hamers, A. S., & Samsing, J. 2019, MNRAS, 487, 5630,\\ndoi: 10.1093/mnras/stz1646\\nHamers, A. S., & Tremaine, S. 2017, AJ, 154, 272,\\ndoi: 10.3847/1538-3881/aa9926\\nHamilton, C., & Modak, S. 2023. https://arxiv.org/abs/2311.04352\\nHamilton, C., & Rafikov, R. R. 2019, ApJL, 881, L13,\\ndoi: 10.3847/2041-8213/ab3468\\nHamilton, C., & Rafikov, R. R. 2019a, Monthly Notices of the\\nRoyal Astronomical Society, 488, 5489,\\ndoi: 10.1093/mnras/stz1730\\n—. 2019b, Monthly Notices of the Royal Astronomical Society,\\n488, 5512, doi: 10.1093/mnras/stz2026\\n—. 2021, MNRAS, 505, 4151, doi: 10.1093/mnras/stab1284\\nHamilton, C., & Rafikov, R. R. 2023, arXiv e-prints,\\narXiv:2306.03703, doi: 10.48550/arXiv.2306.03703\\nHeggie, D. C. 1975, MNRAS, 173, 729,\\ndoi: 10.1093/mnras/173.3.729\\nHeggie, D. C., & Rasio, F. A. 1996, MNRAS, 282, 1064,\\ndoi: 10.1093/mnras/282.3.1064\\nHeisler, J., & Tremaine, S. 1986, Icarus, 65, 13,\\ndoi: https://doi.org/10.1016/0019-1035(86)90060-6\\nHills, J. G. 1975a, AJ, 80, 809, doi: 10.1086/111815\\n—. 1975b, AJ, 80, 1075, doi: 10.1086/111842', metadata={'source': '/content/bm.pdf', 'page': 10}),\n",
              " Document(page_content='12 M. W INTER -GRANI ´C, C. P ETROVICH , & V. P EÑA-DONAIRE\\nHoang, B.-M., Naoz, S., Kocsis, B., Farr, W. M., & McIver, J.\\n2019, The Astrophysical Journal, 875, L31,\\ndoi: 10.3847/2041-8213/ab14f7\\nHoang, B.-M., Naoz, S., Kocsis, B., Rasio, F. A., & Dosopoulou, F.\\n2018, ApJ, 856, 140, doi: 10.3847/1538-4357/aaafce\\nHolman, M., Touma, J., & Tremaine, S. 1997, Nature, 386, 254,\\ndoi: 10.1038/386254a0\\nKocsis, B., & Tremaine, S. 2011, Monthly Notices of the Royal\\nAstronomical Society, 412, 187,\\ndoi: 10.1111/j.1365-2966.2010.17897.x\\nLeigh, N. W. C., Geller, A. M., McKernan, B., et al. 2017, Monthly\\nNotices of the Royal Astronomical Society, 474, 5672,\\ndoi: 10.1093/mnras/stx3134\\nMichaely, E., & Perets, H. B. 2020, MNRAS, 498, 4924,\\ndoi: 10.1093/mnras/staa2720\\nModak, S., & Hamilton, C. 2023, Monthly Notices of the Royal\\nAstronomical Society, 524, 3102, doi: 10.1093/mnras/stad2073\\nNaoz, S. 2016, ARA&A, 54, 441,\\ndoi: 10.1146/annurev-astro-081915-023315Neumayer, N., Walcher, C. J., Andersen, D., et al. 2011, MNRAS,\\n413, 1875, doi: 10.1111/j.1365-2966.2011.18266.x\\nPeters, P. C. 1964, Phys. Rev., 136, B1224,\\ndoi: 10.1103/PhysRev.136.B1224\\nPetrovich, C., & Antonini, F. 2017, The Astrophysical Journal,\\n846, 146, doi: 10.3847/1538-4357/aa8628\\nRasskazov, A., & Rafikov, R. R. 2023, arXiv e-prints,\\narXiv:2310.15374. https://arxiv.org/abs/2310.15374\\nRein, H., & Liu, S.-F. 2012, A&A, 537, A128,\\ndoi: 10.1051/0004-6361/201118085\\nSchödel, R., Feldmeier, A., Kunneriath, D., et al. 2014, A&A, 566,\\nA47, doi: 10.1051/0004-6361/201423481\\nSchödel, R., Gallego-Cano, E., Dong, H., et al. 2018, A&A, 609,\\nA27, doi: 10.1051/0004-6361/201730452\\nStephan, A. P., Naoz, S., Ghez, A. M., et al. 2016, MNRAS, 460,\\n3494, doi: 10.1093/mnras/stw1220\\nStephan, A. P., Naoz, S., Ghez, A. M., et al. 2019, The\\nAstrophysical Journal, 878, 58, doi: 10.3847/1538-4357/ab1e4d\\nTremaine, S. 2023, Dynamics of Planetary Systems\\nTurner, M. L., Côté, P., Ferrarese, L., et al. 2012, ApJS, 203, 5,\\ndoi: 10.1088/0067-0049/203/1/5', metadata={'source': '/content/bm.pdf', 'page': 11}),\n",
              " Document(page_content='BINARY EVOLUTION PERTURBED BY FLYBYS AND TIDAL FIELDS 13\\nAPPENDIX\\nA.IMPULSE APPROXIMATION\\nFrom Collins & Sari (2008), we can write the variation in\\nthe velocity (impulse) ∆⃗ viof one component of the binary\\nwith mass mi(i={1,2}) considering an impact parameter\\n⃗bias follows:\\n∆⃗ vi=2Gmp\\nvpbiˆbi. (A1)\\nThe impact parameter and the velocity of the perturber\\nmust be orthogonal such that ⃗bi·⃗ vp= 0. The trajectory\\nof the perturber is approximated to a straight line, such that\\n⃗ r(t) = ⃗bi+⃗ vpt. The perturber’s velocity is relative to\\nonly one component of the binary; therefore, we can re-\\nlate both impact parameters b1andb2through the equation\\n⃗b2=⃗b1−⃗ r+ ˆvp(⃗ r·ˆvp)considering that ⃗b2must also be\\nperpendicular to ⃗ vp. The overall change in relative velocity\\ncan then be expressed as ∆⃗ v= ∆⃗ v1−∆⃗ v2, where both ∆⃗ v1\\nand∆⃗ v2obey the equation A1.\\nWe can define ∆⃗ vas the sum of the velocity components\\nperpendicular and parallel to the instantaneous orbital veloc-\\nity;∆⃗ v= ∆v⊥+ ∆v∥. We can rewrite it using equation A1\\nas\\n∆⃗ v=2GMp\\nvp ˆb1\\nb1−ˆb2\\nb2!\\n(A2)\\nWhen the perturbing potential is axisymmetric (as is the\\ncase in this scenario), the inner angular momentum and ec-\\ncentricity vectors ⃗jand⃗ emust always be orthogonal such\\nthat⃗j·⃗ e= 0. Therefore, if there is a variation in the eccen-\\ntricity vector due to fly-by encounters, there must also be a\\nvariation in ⃗j. We can write the change in the angular mo-\\nmentum vector as\\n∆⃗j=⃗ r×∆⃗ v√GM binain(A3)\\nwhich, using A1, results in\\n∆⃗j=r\\nG\\nMbinain2Mp\\nvp \\n⃗ r×ˆb1\\nb1−⃗ r×ˆb2\\nb2!\\n. (A4)\\nA.1. Diffusion coefficient analysis\\nIt would be of interest to analyse the relative contribu-\\ntions of the tidal and the impulse elements on the evolu-\\ntion of the angular momentum of the inner binary, for it is\\nan indicator of how likely the binary is to merge at a given\\nmoment. To this purpose, we define a diffusion coefficient\\nD ≡ ⟨∥ ∆j∥2⟩(f,b,i,ω, Ω)/tenc. From equation A4, we can\\nwrite∥∆⃗j∥as\\nFigure A1. Geometry used to calculate ˆr×ˆb1andˆr×ˆb2. The orbit\\nlies on the equatorial plane of a sphere for reference. The flyby\\ntrajectory is approximated to a straight line, and the gray crosses\\nindicate the points where it intersects either the sphere or its equator.\\n∥∆⃗j∥=r\\nG\\nMbinain2Mp\\nvpr\\r\\r\\r\\r\\rˆr×ˆb1\\nb1−ˆr×ˆb2\\nb2\\r\\r\\r\\r\\r,(A5)\\nwhere the term involving the cross products can be evalu-\\nated numerically by averaging over 106different values cal-\\nculated using random impact parameters biand orbital pa-\\nrameters f,ω,iandΩso as to cover all possible stages of\\nthe binary’s orbit. Using this numerical average, we can write\\n⟨∥∆j∥2⟩(f,b,i,ω, Ω)as\\n⟨∥∆⃗j∥2⟩(f,b,i,ω, Ω)≈6.33×10−6\\x12Mbin\\n10M⊙\\x13−1\\x12Mp\\nM⊙\\x132\\n\\x10vp\\n200km/s\\x11−2\\x10ain\\n25au\\x11\\n,\\n(A6)\\nwhere we have used the average ⟨r2⟩=a2\\nin(1 +3\\n2e2) =\\n7\\n4a2\\nin, considering that ⟨e2⟩= 0.5due to the initial thermal\\ndistribution assumed.', metadata={'source': '/content/bm.pdf', 'page': 12}),\n",
              " Document(page_content='14 M. W INTER -GRANI ´C, C. P ETROVICH , & V. P EÑA-DONAIRE\\nFigure A2. Mean square displacement ofq\\n⟨∥∆⃗j∥2⟩as a function\\nof time window width w/τ 0.\\nA.2. Number of flybys\\nThe number of interactions in a secular timescale can be\\ncalculated as τ0/tenc, following equations (4) and (18)\\nτ0\\ntenc≈67.5\\x12Mbin\\n10M⊙\\x131\\n2\\x12MBH\\n4×106M⊙\\x13−1\\n×\\x12aout\\n0.3pc\\x133\\x10ain\\n25au\\x111\\n2\\x12n⋆\\n106pc−3\\x13\\n×\\x10vp\\n200km/s\\x11\\x12f(β)\\n1.8\\x13\\n,(A7)\\nwhere β=s/a out, and\\nf(β) =\\x141\\n2+4(1 + 3 β)\\n(1 +β)3\\x15−1\\n. (A8)\\nA.3. Numerical experiments for R\\nIn order to confirm that the expression derived for Rusing\\nthe impulse approximation is accurate, we simulated a binary\\nthat evolves only by flyby interactions using REBOUND.\\nFrom this numerical experiment, in which we considered\\nMbin= 10M⊙,Mp=M⊙,MBH= 4×106M⊙,ain= 25 au,\\naout= 0.3pc and vp= 200 kms−1, we calculated the mean\\ndisplacement of ∥⃗j∥in different time windows until finding\\na value at which it converges after one secular timescale τ0.\\nThis is shown in figure A2, where we can clearly see how\\nRconverges to ≈0.02after one secular timescale, which\\nmatches the value we retrieve using equation (23) when con-\\nsidering the same initial conditions. In order to keep this\\nsimulation as similar as possble to the derivations using the\\nimpulse approximation, we don’t consider the changes in ain\\nafter each encounter, as this would change the amount of fly-\\nbys per secular timescale.', metadata={'source': '/content/bm.pdf', 'page': 13})]"
            ]
          },
          "execution_count": 6,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "doc=read_doc('/content/')\n",
        "doc"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 7,
      "metadata": {
        "id": "Z9V1ZXz3EgDY"
      },
      "outputs": [],
      "source": [
        "def chunk_data(docs,chunk_size=800,chunk_overlap=50):\n",
        "    text_splitter=RecursiveCharacterTextSplitter(chunk_size=chunk_size,chunk_overlap=chunk_overlap)\n",
        "    doc=text_splitter.split_documents(docs)\n",
        "    return doc"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 8,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "VD9BDUfHEity",
        "outputId": "3d7b15d7-11e9-4ac5-f754-3bd2b8bdb98a"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "73"
            ]
          },
          "execution_count": 8,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "doc=chunk_data(docs=doc)\n",
        "len(doc)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 9,
      "metadata": {
        "id": "vm0ECuV1D4o8"
      },
      "outputs": [],
      "source": [
        "import pinecone\n",
        "from langchain.vectorstores import Pinecone"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 10,
      "metadata": {
        "id": "oC5-a6cMEC52"
      },
      "outputs": [],
      "source": [
        "## Vector Search DB In Pinecone\n",
        "pinecone.init(\n",
        "    api_key=\"_PINECONE_API_KEY_\",\n",
        "    environment=\"_PINECONE_ENVIRONMENT_\"\n",
        ")\n",
        "index_name=\"e5-llama2-rag\""
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 11,
      "metadata": {
        "id": "6oZs1hLUEteD"
      },
      "outputs": [],
      "source": [
        "index=Pinecone.from_documents(doc,embed_model,index_name=index_name)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 12,
      "metadata": {
        "id": "sciEoURl0t4Q"
      },
      "outputs": [],
      "source": [
        "# # If you have already setup and stored vectors in pinecone, you can import index this way.\n",
        "# text_field = \"text\"\n",
        "# index = pinecone.Index(index_name=index_name)\n",
        "# index = Pinecone(\n",
        "#     index, embed_model.embed_query, text_field\n",
        "# )"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 13,
      "metadata": {
        "id": "OyOMBU3uxrMS"
      },
      "outputs": [],
      "source": [
        "def retrieve_query(query,k=2):\n",
        "    matching_results=index.similarity_search(query,k=k)\n",
        "    return matching_results"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 14,
      "metadata": {
        "id": "ar_SdKW9xtcc"
      },
      "outputs": [],
      "source": [
        "from langchain.chains.question_answering import load_qa_chain"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 16,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 129,
          "referenced_widgets": [
            "7de4441f47d84b7095c6f1ce9bae0332",
            "63e8422d85db4bceab5da2ad43e9b1d4",
            "ccc9304861144d07a0b033e14484fd54",
            "f2a8a979dfc34c1392826d5330226932",
            "b400d2a80dec46aab590d9edfee16bc7",
            "637a9aa4fa6f4eb9ab96960066868caa",
            "dfebe8acc442443c85acf1407e3ee076",
            "f24d4c06932f4a579989f57e558b8fa0",
            "99f5274a879649a4bb37175a9b4b2b3f",
            "a614f132f78a495facb5d54b49293778",
            "c41d288018cc48ec8acd7314258b6fd6"
          ]
        },
        "id": "ujYRQBVHytwd",
        "outputId": "7d59aede-b652-40c2-9749-72bad6850321"
      },
      "outputs": [
        {
          "name": "stderr",
          "output_type": "stream",
          "text": [
            "/usr/local/lib/python3.10/dist-packages/transformers/modeling_utils.py:2193: FutureWarning: The `use_auth_token` argument is deprecated and will be removed in v5 of Transformers.\n",
            "  warnings.warn(\n"
          ]
        },
        {
          "data": {
            "application/vnd.jupyter.widget-view+json": {
              "model_id": "7de4441f47d84b7095c6f1ce9bae0332",
              "version_major": 2,
              "version_minor": 0
            },
            "text/plain": [
              "Loading checkpoint shards:   0%|          | 0/2 [00:00<?, ?it/s]"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        },
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Model loaded on cuda:0\n"
          ]
        }
      ],
      "source": [
        "from torch import cuda, bfloat16\n",
        "import transformers\n",
        "\n",
        "model_id = 'meta-llama/Llama-2-7b-chat-hf'\n",
        "\n",
        "device = f'cuda:{cuda.current_device()}' if cuda.is_available() else 'cpu'\n",
        "\n",
        "# set quantization configuration to load large model with less GPU memory\n",
        "# this requires the `bitsandbytes` library\n",
        "bnb_config = transformers.BitsAndBytesConfig(\n",
        "    load_in_4bit=True,\n",
        "    bnb_4bit_quant_type='nf4',\n",
        "    bnb_4bit_use_double_quant=True,\n",
        "    bnb_4bit_compute_dtype=bfloat16\n",
        ")\n",
        "\n",
        "# begin initializing HF items, need auth token for these\n",
        "hf_auth = '_HF_API_KEY_'\n",
        "model_config = transformers.AutoConfig.from_pretrained(\n",
        "    model_id,\n",
        "    use_auth_token=hf_auth\n",
        ")\n",
        "\n",
        "model = transformers.AutoModelForCausalLM.from_pretrained(\n",
        "    model_id,\n",
        "    trust_remote_code=True,\n",
        "    config=model_config,\n",
        "    quantization_config=bnb_config,\n",
        "    device_map='auto',\n",
        "    use_auth_token=hf_auth\n",
        ")\n",
        "model.eval()\n",
        "print(f\"Model loaded on {device}\")"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 17,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "evjouH2fzrll",
        "outputId": "a6e9839c-f71b-4c5a-e100-9ae273038303"
      },
      "outputs": [
        {
          "name": "stderr",
          "output_type": "stream",
          "text": [
            "/usr/local/lib/python3.10/dist-packages/transformers/tokenization_utils_base.py:1714: FutureWarning: The `use_auth_token` argument is deprecated and will be removed in v5 of Transformers.\n",
            "  warnings.warn(\n"
          ]
        }
      ],
      "source": [
        "tokenizer = transformers.AutoTokenizer.from_pretrained(\n",
        "    model_id,\n",
        "    use_auth_token=hf_auth\n",
        ")"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 18,
      "metadata": {
        "id": "bNC9MhwhzzDe"
      },
      "outputs": [],
      "source": [
        "generate_text = transformers.pipeline(\n",
        "    model=model, tokenizer=tokenizer,\n",
        "    return_full_text=True,  # langchain expects the full text\n",
        "    task='text-generation',\n",
        "    # we pass model parameters here too\n",
        "    temperature=0.0,  # 'randomness' of outputs, 0.0 is the min and 1.0 the max\n",
        "    max_new_tokens=512,  # mex number of tokens to generate in the output\n",
        "    repetition_penalty=1.1  # without this output begins repeating\n",
        ")"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 20,
      "metadata": {
        "id": "pM9Re_QD0GbB"
      },
      "outputs": [],
      "source": [
        "from langchain.llms import HuggingFacePipeline\n",
        "\n",
        "llm = HuggingFacePipeline(pipeline=generate_text)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 27,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 176
        },
        "id": "gkBxttWD0KlP",
        "outputId": "4a433104-8be7-410c-a041-eb511723c524"
      },
      "outputs": [
        {
          "data": {
            "application/vnd.google.colaboratory.intrinsic+json": {
              "type": "string"
            },
            "text/plain": [
              "\"?\\n Unterscheidung between different types of dynamical regimes is important for understanding the structure and evolution of the Galactic center. In this section, we will discuss the different types of dynamical regimes that have been identified in the Galactic center using numerical simulations.\\n\\nOne of the earliest studies on the dynamics of the Galactic center was conducted by \\\\citet{1987ApJ...323..654M}. They used a set of simplified models to represent the gravitational potential of the central supermassive black hole (SMBH) and the surrounding stars and gas. They found that the SMBH dominates the gravitational potential in the inner few parsecs of the Galactic center, while the stellar and gas components are more important at larger distances. This study laid the foundation for subsequent investigations into the dynamics of the Galactic center.\\n\\nSince then, numerous simulations have been performed to study the dynamics of the Galactic center. These simulations have revealed several distinct dynamical regimes, including:\\n\\n1. The SMBH-dominated regime: In this regime, the SMBH is the dominant gravitational force in the inner few parsecs of the Galactic center. Stars and gas are strongly affected by the SMBH's gravity, and their orbits are highly eccentric and highly inclined with respect to the disk of the Milky Way.\\n2. The stellar-dominated regime: Outside of the SMBH-dominated region, the stellar component becomes more important. Stars in this region follow more circular orbits, and their orbits are less affected by the SMBH's gravity.\\n3. The gas-dominated regime: In the outermost regions of the Galactic center, the gas component becomes more important. Gas in this region follows highly eccentric orbits, and their orbits are highly inclined with respect to the disk of the Milky Way.\\n4. The mixed regime: In some simulations, a mixed regime has been identified, where both the SMBH and the stellar/gas components play important roles in shaping the dynamics of the Galactic center.\\n\\nThese different dynamical regimes have important implications for our understanding of the structure and evolution of the Galactic center. For example, the SMBH-dominated regime may be responsible for the observed warping and distortion of the Galactic disk, while the stellar\""
            ]
          },
          "execution_count": 27,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "# Check how the model answers without the context of the document.\n",
        "llm(prompt=\"Can you tell me ab0ut Dynamical regimes in the galactic center\")"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 22,
      "metadata": {
        "id": "G-oB82HJ0V3A"
      },
      "outputs": [],
      "source": [
        "chain=load_qa_chain(llm,chain_type=\"stuff\")"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 29,
      "metadata": {
        "id": "A8eDARjD0ano"
      },
      "outputs": [],
      "source": [
        "def retrieve_answers(query):\n",
        "    doc_search=retrieve_query(query)\n",
        "    #print(doc_search)\n",
        "    response=chain.run(input_documents=doc_search,question=query)\n",
        "    return response"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 37,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "caU6U0k50e7W",
        "outputId": "b4adbaf3-d6c9-49cd-cc28-6748e4b25470"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "[' I can certainly try! Based on the provided text, it seems that dynamical regimes in the galactic center refer to the different ways in which binaries move and interact with each other and their surroundings. The text mentions that there are different regimes depending on the distance between the binaries, with tighter binaries experiencing less diffusion in their angular momentum and wider binaries experiencing more diffusion. Additionally, the text notes that the maximum eccentricities achieved by a population of binaries in the galactic center can be affected by their initial conditions, such as their initial eccentricities and inclinations. However, I do not have enough information from the text to provide a detailed answer to your question.']"
            ]
          },
          "execution_count": 37,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "# We can see the difference when rag is used\n",
        "our_query = \"Can you tell me ab0ut Dynamical regimes in the galactic center\"\n",
        "answer = retrieve_answers(our_query)\n",
        "answer.splitlines()"
      ]
    }
  ],
  "metadata": {
    "accelerator": "GPU",
    "colab": {
      "gpuType": "T4",
      "provenance": [],
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