{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "
\n", "\n", "**This is a fixed-text formatted version of a Jupyter notebook**\n", "\n", "- Try online[![Binder](https://static.mybinder.org/badge.svg)](https://mybinder.org/v2/gh/gammapy/gammapy-webpage/v0.19?urlpath=lab/tree/tutorials/analysis/1D/spectrum_simulation.ipynb)\n", "- You may download all the notebooks in the documentation as a\n", "[tar file](../../../_downloads/notebooks-0.19.tar).\n", "- **Source files:**\n", "[spectrum_simulation.ipynb](../../../_static/notebooks/spectrum_simulation.ipynb) |\n", "[spectrum_simulation.py](../../../_static/notebooks/spectrum_simulation.py)\n", "
\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 1D spectrum simulation\n", "\n", "## Prerequisites\n", "\n", "- Knowledge of spectral extraction and datasets used in gammapy, see for instance the [spectral analysis tutorial](spectral_analysis.ipynb)\n", "\n", "## Context\n", "\n", "To simulate a specific observation, it is not always necessary to simulate the full photon list. For many uses cases, simulating directly a reduced binned dataset is enough: the IRFs reduced in the correct geometry are combined with a source model to predict an actual number of counts per bin. The latter is then used to simulate a reduced dataset using Poisson probability distribution.\n", "\n", "This can be done to check the feasibility of a measurement, to test whether fitted parameters really provide a good fit to the data etc.\n", "\n", "Here we will see how to perform a 1D spectral simulation of a CTA observation, in particular, we will generate OFF observations following the template background stored in the CTA IRFs.\n", "\n", "**Objective: simulate a number of spectral ON-OFF observations of a source with a power-law spectral model with CTA using the CTA 1DC response, fit them with the assumed spectral model and check that the distribution of fitted parameters is consistent with the input values.**\n", "\n", "## Proposed approach:\n", "\n", "We will use the following classes:\n", "\n", "* `~gammapy.datasets.SpectrumDatasetOnOff`\n", "* `~gammapy.datasets.SpectrumDataset`\n", "* `~gammapy.irf.load_cta_irfs`\n", "* `~gammapy.modeling.models.PowerLawSpectralModel`" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Setup\n" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "execution": { "iopub.execute_input": "2021-11-22T21:07:13.403788Z", "iopub.status.busy": "2021-11-22T21:07:13.402643Z", "iopub.status.idle": "2021-11-22T21:07:13.571542Z", "shell.execute_reply": "2021-11-22T21:07:13.571858Z" } }, "outputs": [], "source": [ "%matplotlib inline\n", "import matplotlib.pyplot as plt" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "execution": { "iopub.execute_input": "2021-11-22T21:07:13.574481Z", "iopub.status.busy": "2021-11-22T21:07:13.574114Z", "iopub.status.idle": "2021-11-22T21:07:14.145092Z", "shell.execute_reply": "2021-11-22T21:07:14.145293Z" } }, "outputs": [], "source": [ "import numpy as np\n", "import astropy.units as u\n", "from astropy.coordinates import SkyCoord, Angle\n", "from regions import CircleSkyRegion\n", "from gammapy.datasets import SpectrumDatasetOnOff, SpectrumDataset, Datasets\n", "from gammapy.makers import SpectrumDatasetMaker\n", "from gammapy.modeling import Fit\n", "from gammapy.modeling.models import (\n", " PowerLawSpectralModel,\n", " SkyModel,\n", ")\n", "from gammapy.irf import load_cta_irfs\n", "from gammapy.data import Observation\n", "from gammapy.maps import MapAxis, RegionGeom" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Simulation of a single spectrum\n", "\n", "To do a simulation, we need to define the observational parameters like the livetime, the offset, the assumed integration radius, the energy range to perform the simulation for and the choice of spectral model. We then use an in-memory observation which is convolved with the IRFs to get the predicted number of counts. This is Poission fluctuated using the `fake()` to get the simulated counts for each observation. " ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "execution": { "iopub.execute_input": "2021-11-22T21:07:14.150221Z", "iopub.status.busy": "2021-11-22T21:07:14.149839Z", "iopub.status.idle": "2021-11-22T21:07:14.151069Z", "shell.execute_reply": "2021-11-22T21:07:14.151376Z" } }, "outputs": [], "source": [ "# Define simulation parameters parameters\n", "livetime = 1 * u.h\n", "\n", "pointing = SkyCoord(0, 0, unit=\"deg\", frame=\"galactic\")\n", "offset = 0.5 * u.deg\n", "\n", "# Reconstructed and true energy axis\n", "energy_axis = MapAxis.from_edges(\n", " np.logspace(-0.5, 1.0, 10), unit=\"TeV\", name=\"energy\", interp=\"log\"\n", ")\n", "energy_axis_true = MapAxis.from_edges(\n", " np.logspace(-1.2, 2.0, 31), unit=\"TeV\", name=\"energy_true\", interp=\"log\"\n", ")\n", "\n", "on_region_radius = Angle(\"0.11 deg\")\n", "\n", "center = pointing.directional_offset_by(\n", " position_angle=0 * u.deg, separation=offset\n", ")\n", "on_region = CircleSkyRegion(center=center, radius=on_region_radius)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "execution": { "iopub.execute_input": "2021-11-22T21:07:14.156956Z", "iopub.status.busy": "2021-11-22T21:07:14.156626Z", "iopub.status.idle": "2021-11-22T21:07:14.161623Z", "shell.execute_reply": "2021-11-22T21:07:14.161855Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "PowerLawSpectralModel\n", "\n", " type name value unit error min max frozen link\n", "-------- --------- ---------- -------------- --------- --- --- ------ ----\n", "spectral index 3.0000e+00 0.000e+00 nan nan False \n", "spectral amplitude 2.5000e-12 cm-2 s-1 TeV-1 0.000e+00 nan nan False \n", "spectral reference 1.0000e+00 TeV 0.000e+00 nan nan True \n" ] } ], "source": [ "# Define spectral model - a simple Power Law in this case\n", "model_simu = PowerLawSpectralModel(\n", " index=3.0,\n", " amplitude=2.5e-12 * u.Unit(\"cm-2 s-1 TeV-1\"),\n", " reference=1 * u.TeV,\n", ")\n", "print(model_simu)\n", "# we set the sky model used in the dataset\n", "model = SkyModel(spectral_model=model_simu, name=\"source\")" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "execution": { "iopub.execute_input": "2021-11-22T21:07:14.163777Z", "iopub.status.busy": "2021-11-22T21:07:14.163468Z", "iopub.status.idle": "2021-11-22T21:07:14.212561Z", "shell.execute_reply": "2021-11-22T21:07:14.212760Z" } }, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "Invalid unit found in background table! Assuming (s-1 MeV-1 sr-1)\n" ] } ], "source": [ "# Load the IRFs\n", "# In this simulation, we use the CTA-1DC irfs shipped with gammapy.\n", "irfs = load_cta_irfs(\n", " \"$GAMMAPY_DATA/cta-1dc/caldb/data/cta/1dc/bcf/South_z20_50h/irf_file.fits\"\n", ")" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "execution": { "iopub.execute_input": "2021-11-22T21:07:14.218832Z", "iopub.status.busy": "2021-11-22T21:07:14.218523Z", "iopub.status.idle": "2021-11-22T21:07:14.219901Z", "shell.execute_reply": "2021-11-22T21:07:14.220225Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Observation\n", "\n", "\tobs id : 0 \n", " \ttstart : 51544.00\n", "\ttstop : 51544.04\n", "\tduration : 3600.00 s\n", "\tpointing (icrs) : 266.4 deg, -28.9 deg\n", "\n", "\tdeadtime fraction : 0.0%\n", "\n" ] } ], "source": [ "obs = Observation.create(pointing=pointing, livetime=livetime, irfs=irfs)\n", "print(obs)" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "execution": { "iopub.execute_input": "2021-11-22T21:07:14.235579Z", "iopub.status.busy": "2021-11-22T21:07:14.235099Z", "iopub.status.idle": "2021-11-22T21:07:14.319091Z", "shell.execute_reply": "2021-11-22T21:07:14.319291Z" } }, "outputs": [], "source": [ "# Make the SpectrumDataset\n", "geom = RegionGeom.create(region=on_region, axes=[energy_axis])\n", "\n", "dataset_empty = SpectrumDataset.create(\n", " geom=geom, energy_axis_true=energy_axis_true, name=\"obs-0\"\n", ")\n", "maker = SpectrumDatasetMaker(selection=[\"exposure\", \"edisp\", \"background\"])\n", "\n", "dataset = maker.run(dataset_empty, obs)" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "execution": { "iopub.execute_input": "2021-11-22T21:07:14.327848Z", "iopub.status.busy": "2021-11-22T21:07:14.327543Z", "iopub.status.idle": "2021-11-22T21:07:14.328751Z", "shell.execute_reply": "2021-11-22T21:07:14.328939Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "SpectrumDataset\n", "---------------\n", "\n", " Name : obs-0 \n", "\n", " Total counts : 298 \n", " Total background counts : 22.29\n", " Total excess counts : 275.71\n", "\n", " Predicted counts : 303.66\n", " Predicted background counts : 22.29\n", " Predicted excess counts : 281.37\n", "\n", " Exposure min : 2.53e+08 m2 s\n", " Exposure max : 1.77e+10 m2 s\n", "\n", " Number of total bins : 9 \n", " Number of fit bins : 9 \n", "\n", " Fit statistic type : cash\n", " Fit statistic value (-2 log(L)) : -1811.58\n", "\n", " Number of models : 1 \n", " Number of parameters : 3\n", " Number of free parameters : 2\n", "\n", " Component 0: SkyModel\n", " \n", " Name : source\n", " Datasets names : None\n", " Spectral model type : PowerLawSpectralModel\n", " Spatial model type : \n", " Temporal model type : \n", " Parameters:\n", " index : 3.000 +/- 0.00 \n", " amplitude : 2.50e-12 +/- 0.0e+00 1 / (cm2 s TeV)\n", " reference (frozen) : 1.000 TeV \n", " \n", " \n" ] } ], "source": [ "# Set the model on the dataset, and fake\n", "dataset.models = model\n", "dataset.fake(random_state=42)\n", "print(dataset)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You can see that background counts are now simulated" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### On-Off analysis\n", "\n", "To do an on off spectral analysis, which is the usual science case, the standard would be to use `SpectrumDatasetOnOff`, which uses the acceptance to fake off-counts " ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "execution": { "iopub.execute_input": "2021-11-22T21:07:14.331318Z", "iopub.status.busy": "2021-11-22T21:07:14.331017Z", "iopub.status.idle": "2021-11-22T21:07:14.344659Z", "shell.execute_reply": "2021-11-22T21:07:14.344856Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "SpectrumDatasetOnOff\n", "--------------------\n", "\n", " Name : zDADetr1 \n", "\n", " Total counts : 298 \n", " Total background counts : 19.00\n", " Total excess counts : 279.00\n", "\n", " Predicted counts : 300.39\n", " Predicted background counts : 19.02\n", " Predicted excess counts : 281.37\n", "\n", " Exposure min : 2.53e+08 m2 s\n", " Exposure max : 1.77e+10 m2 s\n", "\n", " Number of total bins : 9 \n", " Number of fit bins : 9 \n", "\n", " Fit statistic type : wstat\n", " Fit statistic value (-2 log(L)) : 8.03\n", "\n", " Number of models : 1 \n", " Number of parameters : 3\n", " Number of free parameters : 2\n", "\n", " Component 0: SkyModel\n", " \n", " Name : source\n", " Datasets names : None\n", " Spectral model type : PowerLawSpectralModel\n", " Spatial model type : \n", " Temporal model type : \n", " Parameters:\n", " index : 3.000 +/- 0.00 \n", " amplitude : 2.50e-12 +/- 0.0e+00 1 / (cm2 s TeV)\n", " reference (frozen) : 1.000 TeV \n", " \n", " Total counts_off : 95 \n", " Acceptance : 9 \n", " Acceptance off : 45 \n", "\n" ] } ], "source": [ "dataset_on_off = SpectrumDatasetOnOff.from_spectrum_dataset(\n", " dataset=dataset, acceptance=1, acceptance_off=5\n", ")\n", "dataset_on_off.fake(npred_background=dataset.npred_background())\n", "print(dataset_on_off)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You can see that off counts are now simulated as well. We now simulate several spectra using the same set of observation conditions." ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "execution": { "iopub.execute_input": "2021-11-22T21:07:14.713531Z", "iopub.status.busy": "2021-11-22T21:07:14.633830Z", "iopub.status.idle": "2021-11-22T21:07:14.714818Z", "shell.execute_reply": "2021-11-22T21:07:14.715054Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "CPU times: user 362 ms, sys: 11.4 ms, total: 374 ms\n", "Wall time: 367 ms\n" ] } ], "source": [ "%%time\n", "\n", "n_obs = 100\n", "datasets = Datasets()\n", "\n", "for idx in range(n_obs):\n", " dataset_on_off.fake(\n", " random_state=idx, npred_background=dataset.npred_background()\n", " )\n", " dataset_fake = dataset_on_off.copy(name=f\"obs-{idx}\")\n", " dataset_fake.meta_table[\"OBS_ID\"] = [idx]\n", " datasets.append(dataset_fake)" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "execution": { "iopub.execute_input": "2021-11-22T21:07:15.314641Z", "iopub.status.busy": "2021-11-22T21:07:15.314300Z", "iopub.status.idle": "2021-11-22T21:07:15.316228Z", "shell.execute_reply": "2021-11-22T21:07:15.316411Z" } }, "outputs": [ { "data": { "text/html": [ "
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obs-9530524.400000000000006280.625.03073126049315271.1666666666666771.16666666666667nan252718170.9728768217719697919.5992773600.03600.00.084722222222222230.0067777777777777790.0779444444444444699wstat643.28468794047931229.044.999999999999990.2
obs-9630123.6277.424.96984596942142869.8333333333333469.83333333333334nan252718170.9728768217719697919.5992773600.03600.00.083611111111111110.0065555555555555560.0770555555555555499wstat626.73968424286241189.045.00.2
obs-9729018.8271.225.41798219445482664.0000000000000164.00000000000001nan252718170.9728768217719697919.5992773600.03600.00.080555555555555560.0052222222222222230.0753333333333333499wstat650.9762484493856949.045.00.2
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obs-9932320.8302.226.8570784237687171.1666666666666971.16666666666669nan252718170.9728768217719697919.5992773600.03600.00.089722222222222220.0057777777777777780.0839444444444444599wstat733.15765987680171049.045.00.2
" ], "text/plain": [ "\n", " name counts background excess sqrt_ts ... counts_off acceptance acceptance_off alpha \n", " ... \n", " str7 int64 float64 float64 float64 ... int64 float64 float64 float64 \n", "------- ------ ------------------ ----------------- ------------------ ... ---------- ---------- ------------------ -------------------\n", "stacked 317 18.399999618530273 298.6000061035156 27.082402135529488 ... 92 9.0 45.000000932942285 0.20000000298023224\n", " obs-1 275 22.0 253.0 23.76785365487285 ... 110 9.0 45.0 0.2\n", " obs-2 293 20.6 272.4 25.17110555404655 ... 103 9.0 45.0 0.2\n", " obs-3 280 22.4 257.6 23.982951737405376 ... 112 9.0 45.0 0.2\n", " obs-4 337 20.6 316.4 27.682709945184747 ... 103 9.0 45.0 0.2\n", " obs-5 283 24.400000000000002 258.6 23.727154782347895 ... 122 9.0 44.99999999999999 0.2\n", " obs-6 330 22.400000000000006 307.6 26.889184475727866 ... 112 9.0 44.999999999999986 0.2\n", " obs-7 283 25.8 257.2 23.43087974795853 ... 129 9.0 45.0 0.2\n", " obs-8 308 23.400000000000002 284.6 25.42049273328333 ... 117 9.0 44.99999999999999 0.2\n", " obs-9 299 20.4 278.6 25.57085071486863 ... 102 9.0 45.0 0.2\n", " obs-10 310 15.2 294.8 27.488972161356774 ... 76 9.0 45.0 0.2\n", " obs-11 285 24.000000000000004 261.0 23.933833745454685 ... 120 9.0 44.99999999999999 0.2\n", " obs-12 299 21.000000000000004 278.0 25.4325263895117 ... 105 9.0 44.99999999999999 0.2\n", " obs-13 309 21.6 287.4 25.877406235012618 ... 108 9.0 45.0 0.2\n", " obs-14 320 22.599999999999998 297.4 26.282826028192567 ... 113 9.0 45.00000000000001 0.2\n", " obs-15 283 22.0 261.0 24.25408979304137 ... 110 9.0 45.0 0.2\n", " obs-16 298 24.200000000000003 273.8 24.664218201697352 ... 121 9.0 44.99999999999999 0.2\n", " obs-17 301 28.200000000000003 272.8 24.01180151925227 ... 141 9.0 44.99999999999999 0.2\n", " ... ... ... ... ... ... ... ... ... ...\n", " obs-81 301 19.400000000000002 281.6 25.922347414834622 ... 97 9.0 44.99999999999999 0.2\n", " obs-82 280 22.000000000000004 258.0 24.072587044084237 ... 110 9.0 44.99999999999999 0.2\n", " obs-83 303 22.200000000000003 280.8 25.394928282299105 ... 111 9.0 44.99999999999999 0.2\n", " obs-84 269 21.200000000000003 247.8 23.580247140026987 ... 106 9.0 44.99999999999999 0.2\n", " obs-85 297 22.4 274.6 24.998935845735573 ... 112 9.0 45.0 0.2\n", " obs-86 286 17.800000000000004 268.2 25.423408969515453 ... 89 9.0 44.999999999999986 0.2\n", " obs-87 333 19.8 313.2 27.646851451147217 ... 99 9.0 45.0 0.2\n", " obs-88 315 18.200000000000003 296.8 27.0172059251097 ... 91 9.0 44.99999999999999 0.2\n", " obs-89 287 22.000000000000004 265.0 24.49456558680515 ... 110 9.0 44.99999999999999 0.2\n", " obs-90 286 19.000000000000004 267.0 25.131221887043438 ... 95 9.0 44.99999999999999 0.2\n", " obs-91 285 25.200000000000003 259.8 23.67754591931069 ... 126 9.0 44.99999999999999 0.2\n", " obs-92 313 23.6 289.4 25.664935420194176 ... 118 9.0 45.0 0.2\n", " obs-93 302 18.8 283.2 26.123867522605497 ... 94 9.0 45.0 0.2\n", " obs-94 322 21.8 300.2 26.574029757473202 ... 109 9.0 45.0 0.2\n", " obs-95 305 24.400000000000006 280.6 25.030731260493152 ... 122 9.0 44.99999999999999 0.2\n", " obs-96 301 23.6 277.4 24.969845969421428 ... 118 9.0 45.0 0.2\n", " obs-97 290 18.8 271.2 25.417982194454826 ... 94 9.0 45.0 0.2\n", " obs-98 301 20.400000000000002 280.6 25.687832964675007 ... 102 9.0 44.99999999999999 0.2\n", " obs-99 323 20.8 302.2 26.85707842376871 ... 104 9.0 45.0 0.2" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "table = datasets.info_table()\n", "table" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Before moving on to the fit let's have a look at the simulated observations." ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "execution": { "iopub.execute_input": "2021-11-22T21:07:15.351297Z", "iopub.status.busy": "2021-11-22T21:07:15.350923Z", "iopub.status.idle": "2021-11-22T21:07:15.485791Z", "shell.execute_reply": "2021-11-22T21:07:15.485991Z" } }, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "fix, axes = plt.subplots(1, 3, figsize=(12, 4))\n", "axes[0].hist(table[\"counts\"])\n", "axes[0].set_xlabel(\"Counts\")\n", "axes[1].hist(table[\"counts_off\"])\n", "axes[1].set_xlabel(\"Counts Off\")\n", "axes[2].hist(table[\"excess\"])\n", "axes[2].set_xlabel(\"excess\");" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, we fit each simulated spectrum individually " ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "execution": { "iopub.execute_input": "2021-11-22T21:07:15.490400Z", "iopub.status.busy": "2021-11-22T21:07:15.490083Z", "iopub.status.idle": "2021-11-22T21:07:22.297683Z", "shell.execute_reply": "2021-11-22T21:07:22.297896Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "CPU times: user 6.79 s, sys: 78.8 ms, total: 6.87 s\n", "Wall time: 6.81 s\n" ] } ], "source": [ "%%time\n", "results = []\n", "\n", "fit = Fit()\n", "\n", "for dataset in datasets:\n", " dataset.models = model.copy()\n", " result = fit.optimize(dataset)\n", " results.append(\n", " {\n", " \"index\": result.parameters[\"index\"].value,\n", " \"amplitude\": result.parameters[\"amplitude\"].value,\n", " }\n", " )" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We take a look at the distribution of the fitted indices. This matches very well with the spectrum that we initially injected." ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "execution": { "iopub.execute_input": "2021-11-22T21:07:22.312718Z", "iopub.status.busy": "2021-11-22T21:07:22.312084Z", "iopub.status.idle": "2021-11-22T21:07:22.350008Z", "shell.execute_reply": "2021-11-22T21:07:22.350199Z" }, "nbsphinx-thumbnail": { "tooltip": "Simulate a number of spectral on-off observations of a source with a power-law spectral model using the CTA 1DC response and fit them with the assumed spectral model." } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "index: 3.0036925550380533 += 0.08081469527527077\n" ] }, { "data": { "image/png": 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"text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "index = np.array([_[\"index\"] for _ in results])\n", "plt.hist(index, bins=10, alpha=0.5)\n", "plt.axvline(x=model_simu.parameters[\"index\"].value, color=\"red\")\n", "print(f\"index: {index.mean()} += {index.std()}\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercises\n", "\n", "* Change the observation time to something longer or shorter. Does the observation and spectrum results change as you expected?\n", "* Change the spectral model, e.g. add a cutoff at 5 TeV, or put a steep-spectrum source with spectral index of 4.0\n", "* Simulate spectra with the spectral model we just defined. How much observation duration do you need to get back the injected parameters?" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.9.0" }, "nbsphinx": { "orphan": true } }, "nbformat": 4, "nbformat_minor": 4 }