Orbit of Eberron Around its Sun.
When determining orbital information about a two-body system, the mass of the smaller body is irrelevant if itâs small enough, which is the case when dealing with stars and their planets, or (most) planets and their moons. Unfortunately, we wonât need to use the mass of Eberron in todayâs post, but it will become very important when we start talking about moons.
The formula for determining the period of an orbit is
Where T is the period of one full orbit in seconds, a is the semi-major axis of the ellipse of the orbit in meters, G is the gravitational constant 6.67x10^-11 N*(m/kg)^2, and M is the mass of the larger body in kilograms, in this case the sun.
We know that the full orbit is exactly 336 days. The Eberron Campaign Setting 3.5e describes the Eberron calendar thusly:
âDays are 24 hours long, divided into day and night. Seven days make up a week, four weeks a month, and twelve months a year.â â ECS p. 130
This puts each month at 28 days long, and each year at 336 days long, slightly shorter than an Earth year. This creates a calendar that is far more âperfectâ than any calendar on any planet ever, and feels almost artificial. That said, itâs perfect for a game of D&D where you donât want to bother with keeping track of weeks with ten days and festivals that donât occur on any day of the week but exist outside of the week. (Iâm looking at you, Forgotten Realms. Hissssssss!) More on fantasy calendars, including Eberron (and FR, hisss!) is available here.
The sourcebook says nothing special about Eberronâs sun, so we can assume itâs largely similar to earthâs sun. To differentiate the two, weâll be calling Earthâs sun by its proper name, Sol, and weâll call Eberronâs sun Arrah, named after Dol Arrah, the Sovereign Hostâs sun goddess. (Iâm not using the obvious and delightful pun of calling it Dol, because two other gods have Dol as a first name â or maybe title? Itâs not really clear.)
The mass of Sol is 1.9891x10^30 kg, and so we assume that Arrah has the same mass.
This leaves the only unknown variable in the equation, a. The semi-major axis of an ellipse is a bit like the âlong radius.â More formally, it is half of the major axis, the line segment that runs through both foci to the perimeter of the ellipse. The semi-minor axis, or âshort radius,â is irrelevant in this formula. For purposes of simplicity, weâll assume that Eberron has a perfectly circular orbit. Virtually all orbits are at least a little bit elliptical, but considering how perfect the numbers for the calendar line up, weâre okay with making this assumption.
Image from Wikimedia Commons used under a Creative Commons BY-SA 3.0 licence.
Re-arranging the equation for a gives us:
So we plug in the numbers
This gives us a semi-major axis of 1.4147x10^11m. This is roughly 140 billion meters, 140 million kilometers, 7.8 light minutes, or 0.95 AU, slightly closer to Arrah than Earth is to Sol. Depending on who you ask, this is well within the habitable zone for a Sol-like star.
â´ Eberron orbits its sun at a distance of 1.415x1011 m, or 0.95 AU.