{"title":"Gravitational waves","code":"# /// script\n# requires-python = \">=3.12\"\n# dependencies = [\"marimo>=0.25.1\", \"numpy\", \"matplotlib\"]\n# ///\n\nimport marimo\n\n__generated_with = \"0.25.1\"\napp = marimo.App(width=\"medium\")\n\n\n@app.cell(hide_code=True)\ndef _(mo):\n mo.md(\n r\"\"\"\n # Listen to two black holes merge\n\n On 14 September 2015 the two LIGO detectors recorded a signal that\n rose in pitch and loudness for a fifth of a second and stopped: two\n black holes, about 36 and 29 times the mass of the Sun, spiralling\n into each other 1.3 billion light years away. It was the first direct\n detection of gravitational waves.\n\n Long before the merger, an orbiting pair loses energy to gravitational\n waves at a rate set by one combination of the masses, the chirp mass\n $\\mathcal{M} = (m_1 m_2)^{3/5} / (m_1 + m_2)^{1/5}$. To leading order\n the wave's frequency $\\tau$ seconds before the merger is\n\n $$f(\\tau) = \\frac{1}{\\pi}\\left(\\frac{5}{256\\,\\tau}\\right)^{3/8}\n \\left(\\frac{G\\mathcal{M}}{c^3}\\right)^{-5/8}.$$\n\n Choose the masses and the distance, look at the wave, and play it.\n \"\"\"\n )\n return\n\n\n@app.cell\ndef _():\n import io\n import wave\n\n import marimo as mo\n import numpy as np\n import matplotlib.pyplot as plt\n return io, mo, np, plt, wave\n\n\n@app.cell\ndef _(mo):\n m1 = mo.ui.slider(1.4, 80, step=0.1, value=36, label=\"Mass 1 (Suns)\")\n m2 = mo.ui.slider(1.4, 80, step=0.1, value=29, label=\"Mass 2 (Suns)\")\n distance = mo.ui.slider(40, 3000, step=10, value=410, label=\"Distance (megaparsecs)\")\n mo.vstack([m1, m2, distance])\n return distance, m1, m2\n\n\n@app.cell\ndef _(np):\n G, C = 6.674e-11, 2.998e8\n M_SUN, MPC = 1.989e30, 3.086e22\n\n def chirp(m1, m2, distance_mpc, f_low=20.0, rate=4096):\n \"\"\"Leading-order inspiral from f_low up to the innermost stable orbit.\n\n Returns time (s, zero at the end), strain h(t) for a face-on source,\n frequency f(t), and the stopping frequency.\n \"\"\"\n total = (m1 + m2) * M_SUN\n mc = (m1 * m2) ** 0.6 / (m1 + m2) ** 0.2 * M_SUN\n tc = G * mc / C**3\n f_isco = C**3 / (6**1.5 * np.pi * G * total)\n tau_low = 5 / 256 * tc ** (-5 / 3) * (np.pi * f_low) ** (-8 / 3)\n tau_end = 5 / 256 * tc ** (-5 / 3) * (np.pi * f_isco) ** (-8 / 3)\n tau = np.arange(tau_low, tau_end, -1 / rate)\n f = (5 / (256 * tau)) ** 0.375 * tc ** (-0.625) / np.pi\n phase = -2 * (5 * tc) ** (-0.625) * tau**0.625\n amplitude = 4 / (distance_mpc * MPC) * (G * mc / C**2) ** (5 / 3) * (np.pi * f / C) ** (2 / 3)\n return -(tau - tau_end), amplitude * np.cos(phase), f, f_isco\n\n return (chirp,)\n\n\n@app.cell\ndef _(chirp, distance, m1, m2, mo, np, plt):\n t, h, f, f_isco = chirp(m1.value, m2.value, distance.value)\n cycles = np.sum(f) / 4096\n last = t > -min(0.5, -t.min())\n\n fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(7.2, 5), sharex=True)\n ax1.plot(t[last], h[last] * 1e21, color=\"#0072B2\", lw=1)\n ax1.set_ylabel(\"strain h (10⁻²¹)\")\n ax2.plot(t[last], f[last], color=\"#D55E00\", lw=2)\n ax2.set(xlabel=\"seconds before the end of the inspiral\", ylabel=\"frequency (Hz)\")\n fig.tight_layout()\n\n mo.vstack(\n [\n fig,\n mo.md(\n f\"From 20 Hz, where LIGO starts to hear it, the pair has **{-t.min():.2f} s** and \"\n f\"**{cycles:,.0f} wave cycles** left. The inspiral ends near **{f_isco:.0f} Hz**, \"\n f\"where the orbit can no longer stay stable. The peak strain here is \"\n f\"**{np.abs(h).max():.1e}**: a 4 km detector arm changes length by \"\n f\"{np.abs(h).max() * 4000:.1e} m, about 1/{1.7e-15 / (np.abs(h).max() * 4000):,.0f} \"\n \"of the width of a proton.\"\n ),\n ]\n )\n return h, t\n\n\n@app.cell\ndef _(mo):\n speed = mo.ui.dropdown({\"real time\": 1, \"4 × faster, two octaves higher\": 4}, value=\"real time\", label=\"Playback\")\n speed\n return (speed,)\n\n\n@app.cell\ndef _(h, io, mo, np, speed, wave):\n rate = 4096 * speed.value\n pcm = (h / np.abs(h).max() * 0.9 * 32767).astype(np.int16)\n pcm = np.concatenate([np.zeros(rate // 4, np.int16), pcm, np.zeros(rate // 4, np.int16)])\n buffer = io.BytesIO()\n with wave.open(buffer, \"wb\") as out:\n out.setnchannels(1)\n out.setsampwidth(2)\n out.setframerate(rate)\n out.writeframes(pcm.tobytes())\n mo.vstack(\n [\n mo.audio(buffer.getvalue()),\n mo.md(\n \"The sound is the strain itself, scaled to full volume. Heavy pairs chirp \"\n \"low, below what laptop speakers play well; try the faster playback or \"\n \"two neutron stars (1.4 and 1.4 Suns), whose chirp lasts much longer.\"\n ),\n ]\n )\n return\n\n\n@app.cell(hide_code=True)\ndef _(mo):\n mo.md(\n r\"\"\"\n ---\n **Model and limits.** Leading (quadrupole, Newtonian) order in the\n orbital speed, circular orbits, a face-on source, and no detector\n response. The waveform stops at the innermost stable circular orbit;\n the merger and ringdown that follow need numerical relativity, and the\n real GW150914 signal peaked near 150 Hz. Masses here are as measured\n at the source, without the redshift correction.\n **References.** B. P. Abbott et al. (LIGO and Virgo), *Phys. Rev. Lett.*\n 116 (2016) 061102; P. C. Peters and J. Mathews, *Phys. Rev.* 131 (1963)\n 435; M. Maggiore, *Gravitational Waves*, vol. 1 (2008), ch. 4.\n Fork it to add the next post-Newtonian term or a detector's noise curve.\n \"\"\"\n )\n return\n\n\nif __name__ == \"__main__\":\n app.run()\n","revision":"2026-10-13T02:22:06.505Z","appConfig":{"width":"medium"},"firstPaint":{"notebook":{"cells":[{"code":"mo.md(\n r\"\"\"\n # Listen to two black holes merge\n\n On 14 September 2015 the two LIGO detectors recorded a signal that\n rose in pitch and loudness for a fifth of a second and stopped: two\n black holes, about 36 and 29 times the mass of the Sun, spiralling\n into each other 1.3 billion light years away. It was the first direct\n detection of gravitational waves.\n\n Long before the merger, an orbiting pair loses energy to gravitational\n waves at a rate set by one combination of the masses, the chirp mass\n $\\mathcal{M} = (m_1 m_2)^{3/5} / (m_1 + m_2)^{1/5}$. To leading order\n the wave's frequency $\\tau$ seconds before the merger is\n\n $$f(\\tau) = \\frac{1}{\\pi}\\left(\\frac{5}{256\\,\\tau}\\right)^{3/8}\n \\left(\\frac{G\\mathcal{M}}{c^3}\\right)^{-5/8}.$$\n\n Choose the masses and the distance, look at the wave, and play it.\n \"\"\"\n)","code_hash":"ab1748a76e4a7c526500eb05ae21b1e7","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":true},"id":"Hbol","name":"_"},{"code":"import io\nimport wave\n\nimport marimo as mo\nimport numpy as np\nimport matplotlib.pyplot as plt","code_hash":"504b81865ddf52b1128ae0ec1549c0b6","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":false},"id":"MJUe","name":"_"},{"code":"m1 = mo.ui.slider(1.4, 80, step=0.1, value=36, label=\"Mass 1 (Suns)\")\nm2 = mo.ui.slider(1.4, 80, step=0.1, value=29, label=\"Mass 2 (Suns)\")\ndistance = mo.ui.slider(40, 3000, step=10, value=410, label=\"Distance (megaparsecs)\")\nmo.vstack([m1, m2, distance])","code_hash":"310968c0a1fafb43526780f2c9e1020c","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":false},"id":"vblA","name":"_"},{"code":"G, C = 6.674e-11, 2.998e8\nM_SUN, MPC = 1.989e30, 3.086e22\n\ndef chirp(m1, m2, distance_mpc, f_low=20.0, rate=4096):\n \"\"\"Leading-order inspiral from f_low up to the innermost stable orbit.\n\n Returns time (s, zero at the end), strain h(t) for a face-on source,\n frequency f(t), and the stopping frequency.\n \"\"\"\n total = (m1 + m2) * M_SUN\n mc = (m1 * m2) ** 0.6 / (m1 + m2) ** 0.2 * M_SUN\n tc = G * mc / C**3\n f_isco = C**3 / (6**1.5 * np.pi * G * total)\n tau_low = 5 / 256 * tc ** (-5 / 3) * (np.pi * f_low) ** (-8 / 3)\n tau_end = 5 / 256 * tc ** (-5 / 3) * (np.pi * f_isco) ** (-8 / 3)\n tau = np.arange(tau_low, tau_end, -1 / rate)\n f = (5 / (256 * tau)) ** 0.375 * tc ** (-0.625) / np.pi\n phase = -2 * (5 * tc) ** (-0.625) * tau**0.625\n amplitude = 4 / (distance_mpc * MPC) * (G * mc / C**2) ** (5 / 3) * (np.pi * f / C) ** (2 / 3)\n return -(tau - tau_end), amplitude * np.cos(phase), f, f_isco","code_hash":"78caed4c26fee3ba512b791a1355cb62","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":false},"id":"bkHC","name":"_"},{"code":"t, h, f, f_isco = chirp(m1.value, m2.value, distance.value)\ncycles = np.sum(f) / 4096\nlast = t > -min(0.5, -t.min())\n\nfig, (ax1, ax2) = plt.subplots(2, 1, figsize=(7.2, 5), sharex=True)\nax1.plot(t[last], h[last] * 1e21, color=\"#0072B2\", lw=1)\nax1.set_ylabel(\"strain h (10⁻²¹)\")\nax2.plot(t[last], f[last], color=\"#D55E00\", lw=2)\nax2.set(xlabel=\"seconds before the end of the inspiral\", ylabel=\"frequency (Hz)\")\nfig.tight_layout()\n\nmo.vstack(\n [\n fig,\n mo.md(\n f\"From 20 Hz, where LIGO starts to hear it, the pair has **{-t.min():.2f} s** and \"\n f\"**{cycles:,.0f} wave cycles** left. The inspiral ends near **{f_isco:.0f} Hz**, \"\n f\"where the orbit can no longer stay stable. The peak strain here is \"\n f\"**{np.abs(h).max():.1e}**: a 4 km detector arm changes length by \"\n f\"{np.abs(h).max() * 4000:.1e} m, about 1/{1.7e-15 / (np.abs(h).max() * 4000):,.0f} \"\n \"of the width of a proton.\"\n ),\n ]\n)","code_hash":"82510530e918e19763b779e3d6a3273d","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":false},"id":"lEQa","name":"_"},{"code":"speed = mo.ui.dropdown({\"real time\": 1, \"4 × faster, two octaves higher\": 4}, value=\"real time\", label=\"Playback\")\nspeed","code_hash":"6f9d40052550ca647f447aad7eaccf9a","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":false},"id":"PKri","name":"_"},{"code":"rate = 4096 * speed.value\npcm = (h / np.abs(h).max() * 0.9 * 32767).astype(np.int16)\npcm = np.concatenate([np.zeros(rate // 4, np.int16), pcm, np.zeros(rate // 4, np.int16)])\nbuffer = io.BytesIO()\nwith wave.open(buffer, \"wb\") as out:\n out.setnchannels(1)\n out.setsampwidth(2)\n out.setframerate(rate)\n out.writeframes(pcm.tobytes())\nmo.vstack(\n [\n mo.audio(buffer.getvalue()),\n mo.md(\n \"The sound is the strain itself, scaled to full volume. Heavy pairs chirp \"\n \"low, below what laptop speakers play well; try the faster playback or \"\n \"two neutron stars (1.4 and 1.4 Suns), whose chirp lasts much longer.\"\n ),\n ]\n)","code_hash":"63b1172f73d078a68d10a15ebd75cfe0","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":false},"id":"Xref","name":"_"},{"code":"mo.md(\n r\"\"\"\n ---\n **Model and limits.** Leading (quadrupole, Newtonian) order in the\n orbital speed, circular orbits, a face-on source, and no detector\n response. The waveform stops at the innermost stable circular orbit;\n the merger and ringdown that follow need numerical relativity, and the\n real GW150914 signal peaked near 150 Hz. Masses here are as measured\n at the source, without the redshift correction.\n **References.** B. P. Abbott et al. (LIGO and Virgo), *Phys. Rev. Lett.*\n 116 (2016) 061102; P. C. Peters and J. Mathews, *Phys. Rev.* 131 (1963)\n 435; M. Maggiore, *Gravitational Waves*, vol. 1 (2008), ch. 4.\n Fork it to add the next post-Newtonian term or a detector's noise curve.\n \"\"\"\n)","code_hash":"c2155b20fd5c9ca812a94f7203fe9ebd","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":true},"id":"SFPL","name":"_"}],"metadata":{"marimo_version":"0.25.1"},"version":"1"},"session":{"cells":[{"code_hash":"ab1748a76e4a7c526500eb05ae21b1e7","console":[],"id":"Hbol","outputs":[{"data":{"text/markdown":"
Listen to two black holes merge
\nOn 14 September 2015 the two LIGO detectors recorded a signal that\nrose in pitch and loudness for a fifth of a second and stopped: two\nblack holes, about 36 and 29 times the mass of the Sun, spiralling\ninto each other 1.3 billion light years away. It was the first direct\ndetection of gravitational waves.\nLong before the merger, an orbiting pair loses energy to gravitational\nwaves at a rate set by one combination of the masses, the chirp mass\n||(\\mathcal{M} = (m_1 m_2)^{3/5} / (m_1 + m_2)^{1/5}||). To leading order\nthe wave's frequency ||(\\tau||) seconds before the merger is\n||[f(\\tau) = \\frac{1}{\\pi}\\left(\\frac{5}{256\\,\\tau}\\right)^{3/8}\n\\left(\\frac{G\\mathcal{M}}{c^3}\\right)^{-5/8}.||]Choose the masses and the distance, look at the wave, and play it."},"type":"data"}]},{"code_hash":"504b81865ddf52b1128ae0ec1549c0b6","console":[],"id":"MJUe","outputs":[{"data":{"text/plain":""},"type":"data"}]},{"code_hash":"310968c0a1fafb43526780f2c9e1020c","console":[],"id":"vblA","outputs":[{"data":{"text/html":"
From 20 Hz, where LIGO starts to hear it, the pair has 0.81 s and 23 wave cycles left. The inspiral ends near 68 Hz, where the orbit can no longer stay stable. The peak strain here is 1.2e-21: a 4 km detector arm changes length by 4.8e-18 m, about 1/357 of the width of a proton.
The sound is the strain itself, scaled to full volume. Heavy pairs chirp low, below what laptop speakers play well; try the faster playback or two neutron stars (1.4 and 1.4 Suns), whose chirp lasts much longer.
"},"type":"data"}]},{"code_hash":"c2155b20fd5c9ca812a94f7203fe9ebd","console":[],"id":"SFPL","outputs":[{"data":{"text/markdown":"\nModel and limits. Leading (quadrupole, Newtonian) order in the\norbital speed, circular orbits, a face-on source, and no detector\nresponse. The waveform stops at the innermost stable circular orbit;\nthe merger and ringdown that follow need numerical relativity, and the\nreal GW150914 signal peaked near 150 Hz. Masses here are as measured\nat the source, without the redshift correction.\nReferences. B. P. Abbott et al. (LIGO and Virgo), Phys. Rev. Lett.\n116 (2016) 061102; P. C. Peters and J. Mathews, Phys. Rev. 131 (1963)\n435; M. Maggiore, Gravitational Waves, vol. 1 (2008), ch. 4.\nFork it to add the next post-Newtonian term or a detector's noise curve."},"type":"data"}]}],"metadata":{"marimo_version":"0.25.1","script_metadata_hash":null},"version":"1"},"appConfig":{"sql_output":"auto","width":"medium"},"layout":null,"renderedOn":"browser","formulas":{"||(\\mathcal{M} = (m_1 m_2)^{3/5} / (m_1 + m_2)^{1/5}||)":"M=(m1m2)3/5/(m1+m2)1/5","||(\\tau||)":"τ","||[f(\\tau) = \\frac{1}{\\pi}\\left(\\frac{5}{256\\,\\tau}\\right)^{3/8}\n\\left(\\frac{G\\mathcal{M}}{c^3}\\right)^{-5/8}.||]":"f(τ)=π1(256τ5)3/8(c3GM)−5/8."}},"discussions":{"cells":{},"newHref":"/sciencehub-team/gravitational-waves/discussions/new?notebook=gravitational-waves&version=ver_01M4SH2BM7GZKD1N7GWVT91941","listHref":"/sciencehub-team/gravitational-waves/discussions?notebook=gravitational-waves&version=ver_01M4SH2BM7GZKD1N7GWVT91941"}}