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shot in the dark to fix docs: replacing tau with latex-ified version
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2 changes: 1 addition & 1 deletion docs/faq.rst
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.. toctree::
:maxdepth: 1

faq/Quick-Start.ipynb
faq/Time_Of_Periastron.ipynb



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22 changes: 11 additions & 11 deletions docs/faq/Time_Of_Periastron.ipynb
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"cells": [
{
"source": [
"# τ and Time of Periastron\n",
"# $\\tau$ and Time of Periastron\n",
"\n",
"Here, we will discuss what exactly is τ, the parameter `orbitize!` uses to parametrize the epoch of periastron, and how it is related to other quantities of the epoch of periastron in the literature. \n",
"Here, we will discuss what exactly is $\\tau$, the parameter `orbitize!` uses to parametrize the epoch of periastron, and how it is related to other quantities of the epoch of periastron in the literature. \n",
"\n",
"## Time of Periastron and Motivation for τ\n",
"## Time of Periastron and Motivation for $\\tau$\n",
"\n",
"The time (or epoch) of periastron is an important quantity for describing an orbit. It defines when the two orbiting bodies are closest to one another (i.e., when a planet is closest to its star). In many papers in the literature, the epoch of periastron is described by $t_p$, which is literally a date at which periastron occurs. This is a very important date because we use this date to anchor our orbit in time. \n",
"\n",
"The value of $t_p$ is well constrained when we know we observed periastron, which is often the case for radial velociy or transiting exoplanets when the orbital periods are short and our data covers a full orbital period. In those cases, we know approximately when $t_p$ should be in time, so it is easy to define prior bounds for it. However, in the case of direct imaging, many of our companions have orbital periods that are orders of magnitude larger than the current orbital coverage of the data where we do not really know if the next periastron is in years, decades, centuries, or even millennia. This is the motivation for τ. \n",
"The value of $t_p$ is well constrained when we know we observed periastron, which is often the case for radial velociy or transiting exoplanets when the orbital periods are short and our data covers a full orbital period. In those cases, we know approximately when $t_p$ should be in time, so it is easy to define prior bounds for it. However, in the case of direct imaging, many of our companions have orbital periods that are orders of magnitude larger than the current orbital coverage of the data where we do not really know if the next periastron is in years, decades, centuries, or even millennia. This is the motivation for $\\tau$. \n",
"\n",
"## Definition of τ\n",
"## Definition of $\\tau$\n",
"\n",
"τ is a dimentionless quantity between 0 and 1 defined with respect to a reference epoch $t_{ref}$. For a planet that has a $t_p$ and an orbital period (P), then we define τ as:\n",
"$\\tau$ is a dimentionless quantity between 0 and 1 defined with respect to a reference epoch $t_{ref}$. For a planet that has a $t_p$ and an orbital period (P), then we define $\\tau$ as:\n",
"\n",
"$$\n",
"\\tau = \\frac{t_p - t_{ref}}{P}.\n",
"$$\n",
"\n",
"Because τ is always between 0 and 1, it is easy to figure out the bounds of τ whereas if the orbital period is highly uncertain, it may be difficult to put bounds on $t_p$ that would encompass all allowable bound orbits. \n",
"Because $\\tau$ is always between 0 and 1, it is easy to figure out the bounds of $\\tau$ whereas if the orbital period is highly uncertain, it may be difficult to put bounds on $t_p$ that would encompass all allowable bound orbits. \n",
"\n",
"## Relation to $t_p$\n",
"\n",
"As seen in the above equation, it is relatively straightforward to covert between orbital parameter sets that use τ and $t_p$. You just need to know the orbital period and reference epoch. In `orbitize!`, both the `System` class and the `Results` class has the attribute `tau_ref_epoch` which stores $t_{ref}$, so there should always be a convenient way to grab this number. By default, we use $t_{ref} = 58849$ MJD. \n",
"As seen in the above equation, it is relatively straightforward to covert between orbital parameter sets that use $\\tau$ and $t_p$. You just need to know the orbital period and reference epoch. In `orbitize!`, both the `System` class and the `Results` class has the attribute `tau_ref_epoch` which stores $t_{ref}$, so there should always be a convenient way to grab this number. By default, we use $t_{ref} = 58849$ MJD. \n",
"\n",
"One thing to note that is a given orbit has only a single valid τ, but that an orbit can be defined by many $t_p$, since the orbit is periodic. Thus, $t_p + P$ is another valid time of periastron. \n",
"One thing to note that is a given orbit has only a single valid $\\tau$, but that an orbit can be defined by many $t_p$, since the orbit is periodic. Thus, $t_p + P$ is another valid time of periastron. \n",
"\n",
"We also provide some helper functions to covert between $t_p$ and τ"
"We also provide some helper functions to covert between $t_p$ and $\\tau$"
],
"cell_type": "markdown",
"metadata": {}
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"source": [
"## Relation to Mean Anomaly\n",
"\n",
"The mean anomaly (M) of an orbit describes the current orbital phase of a planet. M goes from 0 to , and M = 0 means the planet is at periastron. Unlike $t_p$ and τ which describe the epoch of periastron, M describes the current position of the planet in its orbit. \n",
"The mean anomaly (M) of an orbit describes the current orbital phase of a planet. M goes from 0 to 2$\\pi$, and M = 0 means the planet is at periastron. Unlike $t_p$ and $\\tau$ which describe the epoch of periastron, M describes the current position of the planet in its orbit. \n",
"\n",
"To compute M of a planet at some time t, we have provided the following helper function:"
],
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