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cf_project_2024_v2

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cf_project_2024_v2

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  1. 02/12/24 Cooling flows: basics Can happen in galaxies, groups of galaxies and galaxy clusters. Consider hot gas inside a dark matter halo. (Hot? What does hot mean?) Initial conditions: hydrostatic equilibrium effect of radiative cooling: inflow → hot gas X1SJSfkjNQII7fknjyRZ+fOeXBenLfP0jFn1LNKfsB5/wDKNqY9</latexit>@" @t= ?neni⇤(T,Z) YJGwSXWDDcCG6lCiEOB9bB7PPDrN6g0T+SF6aXYiqEjecQZGCsFxaofKWC5n4IyHAT1b0BhqrlIZP QS+Z8bQjaU=">AAACKXicbVDLSgMxFM34tr6qLt0Ei1BBy4yIuhGKbly4ULC22CnDnfRODc1khiQj lKG/48ZfcaOgqFt/xLQWfB4IHM65j9wTpoJr47qvztj4xOTU9MxsYW5+YXGpuLxyqZNMMayxRCSqE JftcFGX28oQOYqNtrpAZ0bMEmOL2r5Rdg43O2HALNgTv98l/yeVOxdur7J3vlqpHozhmyBpZJ2Xik 9LNX16SLdlgFQGnPqndn4baPli62ozKJbcijsE/Uu8ESmREc6C4qPfTlgWozRMgNZNz01NKx9sYgL 7BT/TmALrQgeblkqIUbfy4aV9umGVNo0SZZ80dKh+78gh1roXh7YyBnOtf3sD8T+vmZnooJVzmWYG <latexit sha1_base64="sQhNlA+ABxKustTvK + BCG + BH to complicate the picture accretion on BH and feedback heating 1

  2. 02/12/24 accretion on BH and feedback heating: + BCG + BH to complicate the picture Which form of energy? Where? When? How? Cooling flows: basics Can happen in galaxies, groups of galaxies and galaxy clusters. Consider hot gas inside a dark matter halo. eXMJvgB6/MLiwCsGQ==</latexit>3 2 kBTvir µmp ⇡GMtot Hot gas? What does hot mean? <latexit sha1_base64="k+kCILHhCi6xpybR8TB8JgaieD8=">AAACM3icb VDLSsNAFJ34tr6iLt0MFsFVSapUl0UXiiCo2FZoQphMJ3boTDLMTIol5J/c+CMuBHGhiFv/wWmTha8DA4dzzuXOPaFgVGnHebampmdm5+YXFitLyyura/b6Rl slqcSkhROWyJsQKcJoTFqaakZuhCSIh4x0wsHx2O8MiVQ0ia/1SBCfo9uYRhQjbaTAPvMiiXC2l2f1vKCD4AheB5knORxSmeeZx1PIA5FDDwkhkztY5E7geZH SiTapqzywq07NmQD+JW5JqqDERWA/er0Ep5zEGjOkVNd1hPYzJDXFjOQVL1VEIDxAt6RraIw4UX42uTmHO0bpwSiR5sUaTtTvExniSo14aJIc6b767Y3F/7xu qqNDP6OxSDWJcbEoSpk5E44LhD0qCdZsZAjCkpq/QtxHphFtaq6YEtzfJ/8l7XrNbdQal/vV5lFZxwLYAttgF7jgADTBKbgALYDBPXgCr+DNerBerHfro4hOW R ✓ ◆✓2 Mpc ◆ Tvir⇡GM µmp kB ⇡GM µmp kB ⇡GM µmp kB M ⇡ 8 ⇥ 107 K 1015M? 2R 3R R R AAAC4HicjZJLixQxEMfT7WsdX+N69FI4COt 64="jJgj/HMdSKVkAx/25PwWPZ9NP5Q="> <latexit sha1_base yJ43z2apE/PhXOS1Me4tyKkWaTUhaSMwxW 2w80jbyrHxYAbZdxJzrxQshQDlKjEiXWC6V YYbEzqmVpKxQlXx1UVArQwOK1YSyd4KjmQT 1v51mNXUaTiVrgHKrHXmDGjhGK9f95u6B++ yJ43z2apE/PhXOS1Me4tyKkWaTUhaSMwxW 6GH2OvsR5/Cn+Gn9boXG0rnlEzkX8/TfGae uxedvf31ODbIY/KEbJGUbJM98oYckAHh0Sj l6B6VXdjLsh4UZGCVnd2FydikM+mZMEknJN bI</latexit> <latexit sha1_base 64="jJgj/HMdSKVkAx/25PwWPZ9NP5Q="> AAAC4HicjZJLixQxEMfT7WsdX+N69FI4COt l6B6VXdjLsh4UZGCVnd2FydikM+mZMEknJN DuZLgr8CkLI8LwJ1phCOnFli+Lo143Tbrp XLDk2fvHhQxKsfy5ufxKuZB+g+EAsC//zr 64="jJgj/HMdSKVkAx/25PwWPZ9NP5Q="> 2w80jbyrHxYAbZdxJzrxQshQDlKjEiXWC6V XLDk2fvHhQxKsfy5ufxKuZB+g+EAsC//zr AAAC4HicjZJLixQxEMfT7WsdX+N69FI4COt XLDk2fvHhQxKsfy5ufxKuZB+g+EAsC//zr <latexit sha1_base bI</latexit> 6GH2OvsR5/Cn+Gn9boXG0rnlEzkX8/TfGae HkSjQtWnlhGZ+xiRgGWbLQxKhePlADT4Mz uxedvf31ODbIY/KEbJGUbJM98oYckAHh0Sj DuZLgr8CkLI8LwJ1phCOnFli+Lo143Tbrp VyRVSW6V9JgkP6P42vUbN29t3G7duXvv/oP hsK4sEqEpft3Rc2093OdB1IznPqLuYV5VW5 yJ43z2apE/PhXOS1Me4tyKkWaTUhaSMwxW ui3QtOmQdB1n7Bx0bXmlRIlfM+2GaWBzVzK 8Ow82gsELPvqW94sjoQ/4G7Ivc3anaSbLAM ANVIkCYWvFh9Jg1unLAO/2M2rGBhvq5GSK b1Y7qCnRmm3qW7V9FPv9v8t/cDkWphYc0+b 1v51mNXUaTiVrgHKrHXmDGjhGK9f95u6B++ VyRVSW6V9JgkP6P42vUbN29t3G7duXvv/oP VyRVSW6V9JgkP6P42vUbN29t3G7duXvv/oP 2w80jbyrHxYAbZdxJzrxQshQDlKjEiXWC6V hsK4sEqEpft3Rc2093OdB1IznPqLuYV5VW5 yJ43z2apE/PhXOS1Me4tyKkWaTUhaSMwxW 1v51mNXUaTiVrgHKrHXmDGjhGK9f95u6B++ bI</latexit> ANVIkCYWvFh9Jg1unLAO/2M2rGBhvq5GSK 8Ow82gsELPvqW94sjoQ/4G7Ivc3anaSbLAM ui3QtOmQdB1n7Bx0bXmlRIlfM+2GaWBzVzK HkSjQtWnlhGZ+xiRgGWbLQxKhePlADT4Mz 6GH2OvsR5/Cn+Gn9boXG0rnlEzkX8/TfGae 64="jJgj/HMdSKVkAx/25PwWPZ9NP5Q="> uxedvf31ODbIY/KEbJGUbJM98oYckAHh0Sj DuZLgr8CkLI8LwJ1phCOnFli+Lo143Tbrp YYbEzqmVpKxQlXx1UVArQwOK1YSyd4KjmQT HkSjQtWnlhGZ+xiRgGWbLQxKhePlADT4Mz 1v51mNXUaTiVrgHKrHXmDGjhGK9f95u6B++ ui3QtOmQdB1n7Bx0bXmlRIlfM+2GaWBzVzK 8Ow82gsELPvqW94sjoQ/4G7Ivc3anaSbLAM ANVIkCYWvFh9Jg1unLAO/2M2rGBhvq5GSK 2w80jbyrHxYAbZdxJzrxQshQDlKjEiXWC6V VyRVSW6V9JgkP6P42vUbN29t3G7duXvv/oP hsK4sEqEpft3Rc2093OdB1IznPqLuYV5VW5 l6B6VXdjLsh4UZGCVnd2FydikM+mZMEknJN YYbEzqmVpKxQlXx1UVArQwOK1YSyd4KjmQT b1Y7qCnRmm3qW7V9FPv9v8t/cDkWphYc0+b ANVIkCYWvFh9Jg1unLAO/2M2rGBhvq5GSK 8Ow82gsELPvqW94sjoQ/4G7Ivc3anaSbLAM ui3QtOmQdB1n7Bx0bXmlRIlfM+2GaWBzVzK XLDk2fvHhQxKsfy5ufxKuZB+g+EAsC//zr HkSjQtWnlhGZ+xiRgGWbLQxKhePlADT4Mz YYbEzqmVpKxQlXx1UVArQwOK1YSyd4KjmQT hsK4sEqEpft3Rc2093OdB1IznPqLuYV5VW5 uxedvf31ODbIY/KEbJGUbJM98oYckAHh0Sj 6GH2OvsR5/Cn+Gn9boXG0rnlEzkX8/TfGae bI</latexit> b1Y7qCnRmm3qW7V9FPv9v8t/cDkWphYc0+b l6B6VXdjLsh4UZGCVnd2FydikM+mZMEknJN b1Y7qCnRmm3qW7V9FPv9v8t/cDkWphYc0+b <latexit sha1_base AAAC4HicjZJLixQxEMfT7WsdX+N69FI4COt DuZLgr8CkLI8LwJ1phCOnFli+Lo143Tbrp d"/dt=3 " k⇢T d" dt= ?neni⇤(T) tcool= 2 µmpneni⇤(T) latexit> rXevltL1mRmC/2C9f4FjtueeA==</ Zk++S9pH9Wceu34ul5tnE/iWETbaAf J4ZBJoJoPDSFUMrMrpn1iUtImx7IJw ITeRsQBZwJaGmmOdymEkgccOgEg4u H4n9fNdHTq5UykmQZBvz+KMo51gkc jhJpSmg8Vn9O5CRWahgHpjMmuq+mvZ RlN1OQEjogPegaKkgMysvHRxV41ygh b/y4YYJzWIQmnKiVNexU+3lRGpGOR sXpnXQrJ3s+9XqnbNHgP/Jc6EVNEET CNJaB66d0RCqhhPRJGHujg7FD5g4TP R37kDqVgibvQwBS8mPcEiRok2kl85c tIQedoAa6RE3UQhTdo0f0jF6sB+vJe zOzc/MLiUnl5ZXVtvbKx2VZJJim0aM WbSLt1swu5GKCE/wot/xYsHRbx68O V9ehlsQiKWBKp6EUoevHgoYJthSaEz YLn4=">AAACFHicbVBNS8NAEN3U7/p ase64="P2D5JfFGWwwB+mCVie9HRnN <latexit sha1_b a/cVt70OqDgcd7M7szL0g5U9q2P63S GRtaI12wzyc8U87NVBGciuHEvdTm5P+0UYPV6dgLXTcImi8vqhpJ0dB5xLQUFjjKWQCMWxF2pXzCQkgYPqITQkhfPvk1uDzqp8f9b7+Pe+ffV3FskY/kEz kkKTkh5+QnuSBDwskdeYyiKI7uoz/xeryxtMbRauYD+afi3b/hz7Pr</latexit> +93u3v7l840lsOQG2nsdcEcSKFhiAIlXNcWmCokXBXTH3P96hasE0YPcFbDWLEbLSrBGQYq70rMfWYV5cbIlp7RrLKM++yWWaidkEa3vnx2ol9oiU++r6 <latexit sha1_base64="NNQu7qrUoRZ1c1EPCg3KWhur8jk=">AAACVXi cbVFNSxxBEO0ZjR9rEldzzKXJIpjLZsYY9CJIcskhBwO7KuwsQ09PjdtsfwzdNcLSzJ/0Iv4TL4H0foBGU1Dw+r1XdPXropbCYZI8RPHa+puNza3tzs7bd 0/apdwmtmJoYPWZ6qhKq+pziG0oNmvsE3JDgef27zbS/rJouhrkK5Aj6zqIu/eZaXhjQKNXDLnRmlS49gzi4JLaDtZ46BmfMpuYBSgZgrc2C9SaelBYEpa 2

  3. 02/12/24 Cooling flows: basics If tcoolis small enough, radiative cooling becomes important. Small? What does it mean? Several physical timescales control the type of flow: age, tdyn, tsound, tcool Let’s take a closer look Observed cooling times (Voigt & Fabian 2004) Sample of relaxed (cool core) clusters average cluster age (?) z=1 à à t=7.8 Gyr In the center, tcool≪ ≪ age à radiative losses dynamically important àgas loses pressure support in the center and must flow inward this is the “cooling radius” Concept of cooling radius 6 3

  4. 02/12/24 ICM Dynamics: Cooling Flows The character of the flow is determined by the hierarchy of relevant time scales: tcool< tagemeans that radiative losses are important, the ICM can’t be strictly hydrostatic à ICM must flow à cooling flow (CF). tff< tcool means that the gas is ~ hydrostatic. Notice that tff~ tsound. à à the ICM has time to continuously find an equilibrium configuration Not always true. Heating or merging can disrupt equilibrium. Also: in the very center tcooldecreases and we might have tcool~ tff à the gas flows to the center in ~free fall. 7 The free-fall time, aka the dynamical time, is: ✓R ◆1/2 ✓2r 1 1 pG⇢tot tff⇠ ⇠ (GM/R3)1/2⇠ g LQtxkxTHDFesBBsNtSMyIzwW6y+9Opf/PItOGF+guTkg0kGSmec0rAS2lLQWrz3OHEcImTXBNqY2 <latexit sha1_base64="1g5r64kFqOYfzROF sX+y1SZPlPol74K1zASfYAoSNZGs3M07PHWSm4gSh6DsKl5ZXVtcZ688vG5tbX1vbOtSkqTVmPFq f6+AX9XKRhE=">AAACV3icbVFNTxsxFPQuXyEFGuixF6tRpfQSdikIjggOcKkEqAGkbFh5HW9iYX xLXpI1qXKStp2RY0EoyBVQQY/pxVMLAEg2cCuaaSWVYSeg9GbG+p4pIZgZ21ovDP70yxHmh/VGAZ 9AHhNtvtI60yQ==</latexit> +r7CUukMROZ+aQkMDaL3lT8n9evID8aWK7KCpii80V5JXxNeFoyHnLNKIiJJ4Rq7u+K6Zj4TsB/R dOXEC8++TO53uvG+92Dy/328UldRwN9Rz9QB8XoEB2jc3SBeoiif+glWAqWg+fgNVwNG/NoGNQz3 cT86DBniV6XKQ20RJDAc4tpjpnf3av7n7/urPx7p5zc1OwHDrzyJWzI5doPhoDrkNpqx11oxnwZ ◆1/2 tdyn= Indeed, the “official” dynamical time is: g 90xkSGOh5byz+57UTCC46KZdxAkzS6aIgERgiPI4J97hiFMTIEEIVN7diOiAmFTBh5k0I7vzLi6RRLrmV0tltpVC9msWRQ4foCBWRi85RFd2gGqojih7RM3pFb9aT9WK9Wx/T1iVrNnOA/sD6+gHSSp+O</lat <latexit sha1_base64="JUafac1iPABhGWsPZtj/eL2nSL8=">AAACGXicbVDLSsNAFJ34rPUVdel msAh1U5NS0Y1QdOOygn1AU8tkOmmHTiZh5kYoIb/hxl9x40IRl7ryb5w+Ftp64MLhnHu59x4/FlyD43xbS8srq2vruY385tb2zq69t9/QUaIoq9NIRKrlE80El6wOHARrxYqR0Bes6Q+vx37zgSnNI3kHo5h1Q exit> tKXPOCUgJG6tgPd1FMh7o1khi+xJ1gAuOgFitC0jFWW9jNP8f4A8Ml96p6Ws65dcErOBHiRuDNSQDPUuvan14toEjIJVBCt264TQyclCjgVLMt7iWYxoUPSZ21DJQmZ7qSTzzJ8bJQeDiJlSgKeqL8nUhJqPQp Now, assume gas in hydrostatic equilibrium, so that o.o.m.: p R⇠ ?g⇢ ✓?p ◆1/2 ✓p ◆1/2 yT/8K56XJj3FoRUdDP5eZ5IBB6tYZ73q6S5kSGdJ1ljngJeuD1kBtVTI3MBVzsj9AunFeJr82K8rAWmcu31Y+z3brjbgZj0E/kmRCGmSCw279hvUML7TIkSvwvp3EFjslOJRciarGCi8s8Avoi3ag OWjhO+V4hYquBaVHM+PCyZGO1deNErT3Q52GpAYc+PfeSPzMaxeY/e6UMrcFipy/XJQViqKho0lpTzrBUQ0DAe5keCvlAwizYBi+FkZI3n/5IznZbCbbze2jrcbe/mSOefKTrJJ1kpAdskf+kEPSI d1eoe9xIsgl66MyK6F0H0skcFq0JTyztppg0mk5C8s1iG+XNevHvzH3jZw4p41bT24NcDgYfngyRPapX0GMe30dT0zJfZufmvtYVv3xeX6ssrJ94UjosWN8q4sxS8UDIXLZSoxJl1AnSqxGl6cTD <latexit sha1_base64="/dsw0W6lkS0RP6npj4PRSxX4vjE=">AAACRXicbVBNSyMxGM6oq27 pxckTvyn9xH19G/6CF6fIlORZPOD/IG0dMzXWuxow==</latexit> ⇡ cs= ⇢ ⇢ exit> +LOErgCByDU+CDS1AHt6ABmgCDHDyDV/DmPDkvzrvzMWtdcoqZA/AHzucPjJuWYQ==</lat FWZUJKkhAs8WxSmDRsJJGrBPFcGGjS1BWFF7K8RDZBMxNrOyDcGff3mRtM6rfq16cVer1K nJy0GqSYLwCA1I11KBONFhNn0ghydW6cNYKlvCwKn6eyJDXOsxj2wnR2ao572J+J/XTU18 IOnUeYmQglZOOvuHGhiFs/w51/47TNQlsPXDiccy/33hMljGrjed/O0vLK6tp6aaO8ubW9s +vu7be0TBUmTSyZVJ0IacKoIE1DDSOdRBHEI0ba0ehm4rcfidJUigczTkjI0UDQmGJkrNR OAPbiK6ZAajlc=">AAACAHicbVDLSsNAFJ34rPUVdeHCzWAR3FgSqeiy6MZlFfuAJpTJdN <latexit sha1_base64="HpehmjXohKkD8W zD4NYIZwleXafw0BTDs8GgRrKnlvxqt4UcJH4BamAAo2e+xX0JU45EQYzpHXX9xITZkgZih ✓R ◆1/2 tsc=R R R The sound crossing time can be written as: ⇠ (p/⇢)1/2⇠ (gR)1/2= = t↵ cs g JOoiiN2vZ2rA2rXd73d6ydz6ttlX2rKNfZe99AFsLuOo=</latexit> <latexit sha1_base64="2jxhB92cSpn3MI dC8fZ1XgPcP0cZgoc2LAY/ZnR06k1kMZGKckMNB/tRH5n9bNIDzu5TxOM2Ax/VwUZgJDgkf 5d4wLPxa3Wk648KTwC1BHZV16ddevH5CM8lioIJo3XWdFHo5UcCpYEXVyzRLCX0kEesaGBPJ eRMvEhoKwYn/g=">AAACa3icbVHLSsNAFJ3EV62vqgtFXQwWoW7aRCq6EUQ3LlWsCk0Nk+kk OskUZR2aiETdB0QzwWPWAQ6C3aeKERkIdhc8no/0uyemNE/iGximrCdJFPOQUwKG8mvP4Oee HZxJwsyNUEI2fqI7/8CN/+C0Bl/1wsDhnHPvnTkTpIJrcJxXy56anpmdq8xXFxaXlldqq2u3 B4z5XjIIYGkCo4uaumA6ICQjM91RNCO7fJ0+C24Om224eXrXrp2dlHBW0jXZRA7noCJ2iC3S kljTAp9gL1SE5tdFTn1dYE9z+U010panBsn+Q+62DooJNbr+UswcwULAjS81KjzFowHg0mMs 8 4

  5. 02/12/24 ICM Dynamics: Cooling Flows Hydro equations governing CFs: stellar mass loss mass model (DM, BCG…) ? radiative cooling stellar (SNIa) heating! specific thermal energy stellar source term often negligible in cluster cooling flows (convince yourself) generic heating (which one?) 9 Cooling Rate (a key number): the easy way So, we have seen that gas wants to cool down to low temperature. How much? This is correct for a perfect steady-state, but is a good approximation for real objects too. Typically, for a massive cluster, Lcool~ 1045erg/s, T ~ 108K à Ṁcool~ 500 M¤/yr. ß This is the key point This is a lot! Where is the cold gas located? Which physical properties should it have? Which consequences for BCG evolution? Does cold gas feed the BH? Does it form new stars? 10 5

  6. 02/12/24 The previous formula is easy to understand. 3 kBT One gram of gas has energy that can be radiated away. 2 µmp Erad=3 kBT µmpM i5L3bis0Bu0IUymk3boTBJmJkIJ2brxVdy4UMStb+DOt3HaZqGtPwx8/Occzpw/SDhT2nG+rbX1jc2t7dJOeXdv/+DQPjruqDiVhLZJzGPZC7CinEW0rZnmtJdIikXAaTeY3M7q3QcqFYujlp4m1BN4FLGQEayN5dtoEEpMsss8q+ULnPg N1MqzgUiR8JPctytO1ZkLrYJbQAUKNX37azCMSSpopAnHSvVdJ9FehqVmhNO8PEgVTTCZ4BHtG4ywoMrL5pfk6Nw4QxTG0rxIo7n7eyLDQqmpCEynwHqslmsz879aP9XhjZexKEk1jchiUZhypGM0iwUNmaRE86kBTCQzf0VkjE0e2oRX NiG4yyevQqdWdZ2qe39VqTeKOEpwCmdwAS5cQx3uoAltIPAIz/AKb9aT9WK9Wx+L1jWrmDmBP7I+fwCN75o6</latexit> NiG4yyevQqdWdZ2qe39VqTeKOEpwCmdwAS5cQx3uoAltIPAIz/AKb9aT9WK9Wx+L1jWrmDmBP7I+fwCN75o6</latexit> N1MqzgUiR8JPctytO1ZkLrYJbQAUKNX37azCMSSpopAnHSvVdJ9FehqVmhNO8PEgVTTCZ4BHtG4ywoMrL5pfk6Nw4QxTG0rxIo7n7eyLDQqmpCEynwHqslmsz879aP9XhjZexKEk1jchiUZhypGM0iwUNmaRE86kBTCQzf0VkjE0e2oRX i5L3bis0Bu0IUymk3boTBJmJkIJ2brxVdy4UMStb+DOt3HaZqGtPwx8/Occzpw/SDhT2nG+rbX1jc2t7dJOeXdv/+DQPjruqDiVhLZJzGPZC7CinEW0rZnmtJdIikXAaTeY3M7q3QcqFYujlp4m1BN4FLGQEayN5dtoEEpMsss8q+ULnPg <latexit sha1_base64="PdHnG1M4CodSSoxD2zrz9joCbKk=">AAACCXicbZDLSsNAFIZPvNZ6i7p0M1gEVyWpg NiG4yyevQqdWdZ2qe39VqTeKOEpwCmdwAS5cQx3uoAltIPAIz/AKb9aT9WK9Wx+L1jWrmDmBP7I+fwCN75o6</latexit> <latexit sha1_base64="PdHnG1M4CodSSoxD2zrz9joCbKk=">AAACCXicbZDLSsNAFIZPvNZ6i7p0M1gEVyWpg N1MqzgUiR8JPctytO1ZkLrYJbQAUKNX37azCMSSpopAnHSvVdJ9FehqVmhNO8PEgVTTCZ4BHtG4ywoMrL5pfk6Nw4QxTG0rxIo7n7eyLDQqmpCEynwHqslmsz879aP9XhjZexKEk1jchiUZhypGM0iwUNmaRE86kBTCQzf0VkjE0e2oRX <latexit sha1_base64="PdHnG1M4CodSSoxD2zrz9joCbKk=">AAACCXicbZDLSsNAFIZPvNZ6i7p0M1gEVyWpg NiG4yyevQqdWdZ2qe39VqTeKOEpwCmdwAS5cQx3uoAltIPAIz/AKb9aT9WK9Wx+L1jWrmDmBP7I+fwCN75o6</latexit> N1MqzgUiR8JPctytO1ZkLrYJbQAUKNX37azCMSSpopAnHSvVdJ9FehqVmhNO8PEgVTTCZ4BHtG4ywoMrL5pfk6Nw4QxTG0rxIo7n7eyLDQqmpCEynwHqslmsz879aP9XhjZexKEk1jchiUZhypGM0iwUNmaRE86kBTCQzf0VkjE0e2oRX i5L3bis0Bu0IUymk3boTBJmJkIJ2brxVdy4UMStb+DOt3HaZqGtPwx8/Occzpw/SDhT2nG+rbX1jc2t7dJOeXdv/+DQPjruqDiVhLZJzGPZC7CinEW0rZnmtJdIikXAaTeY3M7q3QcqFYujlp4m1BN4FLGQEayN5dtoEEpMsss8q+ULnPg <latexit sha1_base64="PdHnG1M4CodSSoxD2zrz9joCbKk=">AAACCXicbZDLSsNAFIZPvNZ6i7p0M1gEVyWpg i5L3bis0Bu0IUymk3boTBJmJkIJ2brxVdy4UMStb+DOt3HaZqGtPwx8/Occzpw/SDhT2nG+rbX1jc2t7dJOeXdv/+DQPjruqDiVhLZJzGPZC7CinEW0rZnmtJdIikXAaTeY3M7q3QcqFYujlp4m1BN4FLGQEayN5dtoEEpMsss8q+ULnPg lJLepGKBXBjVChN2hCmEwn7dCZJMxMhBLyEm58FTcuFHEruPNtnLZZaOsPAx//OYcz5/djRqWyrG9jaXlldW29sFHc3Nre2TX39tsySgQmLRyxSHR9JAmjIWkpqhjpxoIg7jPS8UfXk3rngQhJo7CpxjFxORqENKAYKW155umNlw <latexit sha1_base64="muu2MVnHCbc6HrEpskb6qO00jeU=">AAACFXicbZDLSsNAFIYnXmu9RV26GSyCCy iJCPapIFixsQaEBdV/hXiIdCJKB1nUIdjzJy9Cu1K2z8vV+2qpVs/jKIBDcAROgA0uQA3cggZoAQwewTN4BW/Gk/FivBsfs9YlI585AH9kfP4AlY6eeA==</latexit> rUz+AVdAKBcHqWpZVshiOvDptZ6vAEci/O4J1nlqyyNRVcBDuHEsjV8Mwvpx/hhJNQYYak7NlWrNwUCUUxI1nRSSSJER6hAelpDBEn0k2nV2XwWDt9GERCv1DBqft7IkVcyjH3dSdHaijnaxPzv1ovUcGlm9IwThQJ8WxRkDCoIj If a mass M cools, it radiates all its thermal energy à 2 Then if the gas cools at a rate Ṁcoolthe power emitted (luminosity) is Lx=3kBT ˙Mcool 2µmp O863tbS8srq2ntvIb25t7+zae/tNJRKJSQMLJmQ7QIowGpGGppqRdiwJ4gEjrW G4v/eZ1EhxfdlEZxokmEpx+FCYNawHFMsEclwZqNDEFYUrMrxANkQtImzLwJwZ B4NfZb90QqKqK6HsWky1E/oiHFSBvJt50b/wFeQi+UCKencOhXYT1LSx5PIPfj DHo9oeGtn3qSQywEy3y74BSdCeAicWekAGao+faneQMnnEQaM6RUx3Vi3U2R1B NCoUJf0ClDJs20oclkSDJiGeY33Pgrblwo4lJX/o3pY6GtBy4czrk3ufcEMaNK <latexit sha1_base64="gdj/b1DjDZ cz8SCwYGZhuLiTi3M=">AAACGXicbVDLSgMxFM34rPU16tJNsAiuykyt6EYode 0/eZE0S0W3XDy7Kxcq1VkcOXAIjsAJcME5qIBrUAMNgMEjeAav4M16sl6sd+tj2 rpkzWYOwB9YXz9Jl59J</latexit> QzkuW9RJEY4SHqk46hEeJEddPJZRk8NkoPhkKaijScqL8nUsSVGvHAdHKkB2re So, why we have “5/2” in the formula of the previous slide? The reason is the same as the “5/2” in the cooling time! The gas cooling at constant pressure radiates its enthalpy, not energy. 11 Add radiative losses as an additional source term: See below for a fit for Λ(T). Do not allow the temperature to go below 104K (gas is assumed to be photoionized). Thus, the logical steps are: - calculate the new e(i) updated for radiative cooling; - calculate the new temperature, T(i)=e(i)/(cv*d(i)) - if T < 104K set T = 104K - recalculate - set again the boundary conditions. Qej28xvj4lUVPCGnsTEi9CA05BipI3k28ekTM/hDcT+GPbkUMDs2TCXb5ecijMDXCZuTkogR923v3p9gZOIcI0Z <latexit sha1_base64="HrenJb9tty7SY/7XRmTF2V 7Boc0=">AAACAXicbVDLSgMxFM34rPU16kZwEyxC3ZQZKepGKIjgskJf0A5DJs20oZlkSDKFMtSNv+LGhSJu/Qt 3/o2ZdhbaeiDh5Jx7ubkniBlV2nG+rZXVtfWNzcJWcXtnd2/fPjhsKZFITJpYMCE7AVKEUU6ammpGOrEkKAoYa p5XGiCcfzQWHCoBYwiwP2qSRYs4khCEtq/grxEEmEtQmtaEJwF1deJq2LintZqT5US7W7PI4COAGnoAxccAVq4B UqrrOrH2UiQ1xYxMi71EkRjhERqQrqEcRUR56WyDKTwzSh+GQprDNZypvztSFCk1iQJTGSE9VIteJv7ndRMdXns 7UQRNg8AiewSt4s56sF+vd+piXrlh5zxH4A+vzB1n2lEg=</latexit>e(i) = cv⇢(i)T(i) 6

  7. 02/12/24 Grid in spherical coordinates In order to simulate the evolution of a SNR, we need to modify the hydrocode somewhat. Of course we use spherical coordinates. GRID: The grid should extend from 0 to ~ 70 pc (or more). Use an uniformly spaced grid, with N ~ 500 points (or more). Use spherical coordinates. I suggest to make the first grid point negative, and the “zero” to correspond to the second point. That is: xa(1)=-Δx; xa(2)=0; xa(3)=Δx; xa(3)=2Δx; etc. Of course, a negative value for xa(1) does not make sense, but it is used only for boundary conditions and we can ignore this apparent inconsistency. ● ● ● ● xa(1) xa(3) xa(2)=0 A fit of the cooling function to use in the code (red line). (Sharma et al., 2010, ApJ, 720, 652) (0.02 keV = 2.32e5 K) Note: 1 keV = 1.16×107K Note: I made a better fit than this! It is now included in the cf.f90 code 7

  8. 02/12/24 Fit of the cooling function by Sharma et al., 2010, ApJ, 720, 652 (red line). Now, however, the code uses the fit represented with the black solid line. Numerical exercise: the code cf.f90 The program cf.f90 solves the hydro equations for a standard CF model (no heating, no thermal conduction). The only physics used is hydro + gravity + radiative cooling. The code is basically the same as the one you used in the CAS class with the addition of gravity in the momentum equation + a better cooling function L(T,Z). This version includes stellar sources (winds + SNIa) and a tracer equation representing the Fe density (no turbulent diffusion for that, though). The mass model adopted is the same as you used in I project for the CAS class: DM halo + BCG + BH. As I.C. the ICM is initially in hydrostatic equilibrium, with the initial temperature profile you used in the CAS class. The initial baryon fraction is 0.16. The numerical grid is NOT uniform, but more refined at the center. The first grid spacing is set to 50 pc and it extends to 9 Mpc (> Rvir~ 3 Mpc). It uses 1000 grid points. 8

  9. 02/12/24 Mass model for a massive cluster Mgas(initial) Mtot M* MICM(20 kpc) ~ 1011M⊙> Mlost MDM BH 20 kpc 17 The code writes a lot of stuff, but most important are the files clu0001, clu0002, clu0003, clu0004, clu0005, which contain the hydro variables at the evolutionary times 1, 2, 3, 4, 5 Gyr, respectively. It also writes the mass of hot and cold gas, the cooling rate mdot, the X-ray surface brightness, tcooland tdyn, and more. What to do? Must do things: 1) Uderstand the code. Check if everything is ok. 2) Make plots of variable at time 1, 2, 3, 4 and 5 Gyr. 3) Calculate and plot le X-ray luminosity vs radius radius at those times 4) Calculate and plot the cooling rate vs. time and compare with the analytic formula 9

  10. 02/12/24 What to do? Optional calculations: 1) Plot the Fe abundance vs radius. Change the SNIa rate and see if you get a Zfeprofile in agreement with observation (e.g. the Rebusco profile used in the first project) 2) You can ‘‘play’’ with parameters: for instance, increase the density (by varying the parameter den0, the assumed central gas density). Discuss how the character of the flow changes. 3) Calculate cooling flows in galaxies, like the MW or an E galaxy. 4) CalculateAGN feedback models, both radiative or mechanical heating (several interesting problems with no clear answer can be investigated, more will be done in Galaxy Cluster class). How to calculate CF models for other systems? In order to do that you have to change some parameters of the simulations: 10

  11. 02/12/24 For a galaxy group, the model parameters can be changed as: For a galaxy group, the model parameters can be changed as: mvir = 4.75×1013M⨀ rvir = 928 kpc rs = 106.96 kpc conc = 8.68 conc = 8.68 mvir = 4.75×1013M⨀ rvir = 928 kpc rs = 106.96 kpc The central galaxy can be described by: The central galaxy can be described by: mstars = 6×1011M⨀ reff = 8 kpc reff = 8 kpc mstars = 6×1011M⨀ For the SMBH you can keep the mass (109M) of the BCG in cluster. For the SMBH you can keep the mass (109M) of the BCG in cluster. The initial gas can be assumed isothermal (for now), with T = 1.49×107K and you have to find the central density den0 in order to have a baryon fraction ~ 0.16. For an isolated massive elliptical galaxy the model parameters can be changed as: mvir = 5×1012M⨀ rvir = 438 kpc rs = 41.14 kpc conc = 10.65 The central galaxy can be described by: mstars = 6×1011M⨀ reff = 8 kpc Again, for the SMBH you can keep the mass (109M) of the BCG in cluster. The initial gas can be assumed isothermal, with T = 4.41×106K and you have to find the central density den0 in order to have a baryon fraction ~ 0.16. 11

  12. 02/12/24 General method to set up I.C. for any mass system 1) decide the virial mass. 2) NFW mass MNFW= (1 – fbar)×Mvir (this is just a detail…) 3) The virial radius rvir: ✓ ~70 km/s/Mpc (but use cgs units here!) ◆1/3 <latexit sha1_base64="lKl9oMLKhz2oy4ZjLP i87DUSWrs=">AAACSXicbZBLSwMxFIUzrc/6qrp0EyyCbuqMFnUjFHXhRlCwKnRqzaR32mDmQXJHKM /C9rJujvtVIRxglCyIcP+YmkGNEcK+0IBRxl3wiWcxCc8h4zENHALxkIzu8v/xUXW1Vnp1o7q1XqByM P8PTfu3Pkf3LhQxJWZtuDzQuDknHsI+bxYCo22/WQVimPjE5NT06WZ2bn5hfLi0oWOEsWhwSMZqSuPa TbXrylWZpzY0FdY9AIvtWclUvGl65Epjlhuj2kG5cp87mdtYuV+yqPRj6VzgjUSGjOW2XH91OxJMAQu SSad107BhbKVMouISs5CYaYsZvWReaRoYsAN1KByQyumacDvUjZU6IdOB+b6Qs0LofeGYzYNjTv7Pc cU2SFrJJ14pBdUifH5JQ0CCf35Jm8kjfrwXqx3q2P4WrBGnWWyY8pFD8Bjh6yKA==</latexit> ZAihAYKlHAVK2CBJ+HSuz3M88s7UFpE4Tn2Y2gFrBsKX3CGxmqXb1Q7dVVA74TK6D51JfhI16nrK8b 3Mvir 4⇡?vir⇢crit Dvir~ 103, rcrit= 3H02/(8pG) rvir= 4) The concentration, c = rvir/rs, defines the scale radius rs. There is a relation between c and Mvir(ex. Dutton & Macciò, 2014): (here h ~ 0.7) 5) Initial gas temperature: the simplest assumption is isothermal gas, with T given by the M-T relation (ex Shimizu et al. 2003, Arnaud et al., 2005, Babyk & McNamara 2023): T0= (Shimizu et al. 2003) The central gas density can be chosen by requiring that fbar~ 0.16, or similar values when appropriate (for galaxies, for instance, it could be fbar~ 0.1…). M–T relation (Babyk & McNamara 2023) 12

  13. 02/12/24 eoc + stellar source term Fe eoc + source term stellar Fe abundance ZFe= abundance of the SNIa ejecta Specific rates of stellar mass return and SNIa mass return These define the T for stellar wind material and SNIa material average T for all the gas ejected by stars & SNIa eoe + stellar source term 13

  14. 02/12/24 !! tshift = 8.7*1.e9*yr 14

  15. 02/12/24 Test calculation: is hydrostatic equilibrium ok? 1) Run the code without radiative cooling for 1 Gyr (or more). Nothing should change! 2) Check the energy conservation: Ekin, Ethermaland Epotshould be constant in time 3) Check the energy conservation with radiative cooling on. CF standard + stellar terms SNu = 0.2 ZFe,*= 0.8 solar Initial ZFe,ICM= 0.3 solar t = 1,2,3,4,5 Gyr Evolve from 8.7 to 13.7 Gyr tcool unphysical 15

  16. 02/12/24 X-ray surface brightness Classical peaked SX Fe abundance evolution Central peak almost not present 16

  17. 02/12/24 On a better scale… LXevolution standard CF 17

  18. 02/12/24 Hot and cold gas mass vs. time Hot and cold gas mass evolution standard CF Cooling rate Standard CF 18

  19. 02/12/24 Gravitational potentials for NFW DM halo and the BCG (BCG) (the quantity f(c) is calculated in the code and it is named fc) Comparison between gravity from mass and gravity from potential (see formulae above) This is just a check! 19

  20. 02/12/24 Energies for a pure CF model Ekin Eth Epot Etot Check: Hydrostatic equilibrium ok? à yes velocity ~ 0, other profiles unchanged. 20

  21. 02/12/24 Varying the SNIa rate Same calculation, but with SNu = 0.8 (large!) Almost the same as SNu = 0.2 à SN energy not important Iron peak still weak 21

  22. 02/12/24 Same calculation, but with SNu = 3 (huge!) Again, ~ the same as SNu = 0.2 àSN energy not important à(mmh…) Iron peak present, but not strong 22

  23. 02/12/24 Even with SNu = 10 (crazy) the dynamics changes only in the very center à CF! Let’s go crazy: SNu = 100! Outflow! First example of feedbak Crazy abundance… 23

  24. 02/12/24 AGN heating in cluster CFs Compton (radiative heating) Heating rate per unit volume is: So the parameters to choose are: TC, the Compton temperature of the AGN radiation, and LAGN, the radiative power by the AGN. A physically sound model should have a prescription to activate the AGN and to link LAGNto the accretion rate of the central BH. However, let us discuss first simple continuous heating models (LAGN= constant). Also, we adopt TC= 2×108K (a factor of ~ 10 higher than typical values for QSO). Comparison of 3 models: LAGN= 0, 1, 10 × LEdd. Here MBH= 109M⨀ à Ledd= 1.3×1047erg/s. Also: SNu(tnow) = 0.2. Note: the LAGN= 0 model is the same of pure CF model discussed earlier. 24

  25. 02/12/24 Log ne LAGN= 0 (usual standard CF) red blue green = 3 Gyr yellow = 4 Gyr black = 5 Gyr = 1 Gyr = 2 Gyr tcool LAGN= LEdd (heated model) red blue green = 3 Gyr yellow = 4 Gyr black = 5 Gyr = 1 Gyr = 2 Gyr 25

  26. 02/12/24 Cooling rate LAGN= LEdd Hot and cold gas mass vs. time LAGN= LEdd 26

  27. 02/12/24 X-ray luminosity vs time LAGN= LEdd Iron abundance profiles LAGN= LEdd 27

  28. 02/12/24 LAGN= 10×Ledd (huge feedback!) red blue green = 3 Gyr yellow = 4 Gyr black = 5 Gyr = 1 Gyr = 2 Gyr Cooling rate LAGN= 10×Ledd 3 episodes of (strong) cooling Where do the gas cool? 28

  29. 02/12/24 Hot and cold gas mass vs. time LAGN= 10×Ledd almost the same gas mass cools! X-ray luminosity LAGN= 10×Ledd 29

  30. 02/12/24 Iron abundance profiles LAGN= 10×Ledd Abundance profile rules out this model Thermal feedback: thermal energy source in the center The above discussed Compton (radiative) heating, though possibly important for some systems, does not represent the dominant feedback channel for clusters and massive galaxies. Likely, a combination of relativisic jets and non-relativistic AGN outflows control and regulate the evolution of ICM/IGM/ISM. A rough, but nevertheless revealing, way to simulate a general feedback is to inject thermal energy in the central region of clusters/galaxies, with a given power, timescale, etc. How to do that? Use source terms for energy and mass in the hydro equations: edot and mdot, ≠ 0 only for 0 < t < tAGN. How to define edot and mdot? 30

  31. 02/12/24 s = feedback scalelength 31

  32. 02/12/24 Here: EAGN= 1060erg tAGN= 107yr LAGN= 3.16×1045erg/s s = 2 kpc with cooling red blue green = 60 Myr yellow = 80 Myr black = 100 Myr = 20 Myr = 40 Myr à early evolution! 32

  33. 02/12/24 X-ray surface brightness red blue green = 60 Myr yellow = 80 Myr black = 100 Myr = 20 Myr = 40 Myr no SXpeak Same model: CF re-establishes itself EAGN= 1060erg tAGN= 107yr LAGN= 3.16×1045erg/s s = 2 kpc with cooling red blue green = 600 Myr yellow = 800 Myr black = 1 Gyr = 200 Myr = 400 Myr à 1 Gyr evolution! 33

  34. 02/12/24 Same model, without cooling red blue green = 60 Myr yellow = 80 Myr black = 100 Myr = 20 Myr = 40 Myr Energy budget Etot Eth Epot Ekin Ugh! 34

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