All Experts Redux: Correcting The Orbital Period For A Fictional Planet (In A Scifi Novel)

 Question: 

I am writing a science fiction novel set on an alien planet in another solar system.  The orbital period of my planet is 438 Earth days or 10512 hours. Since there are 36 hours in one of my planet's days then there are only 292 planet days in its year. 

By using 292/365 or 0.8 years, in my calculation, I came up with a smaller semimajor axis (7/8 AU). By using 438 Earth days, I arrive at 1.129 AU. How will this affect the (axial) tilt I need, say of 11 degrees? 

Answer:

First, there is only ONE correct orbital period, not two. The error you made here was in confusing the assigned actual 'Earth days' for your planet's orbit (which is the actual parameter to be used in the Kepler eqn.) with an artificial value you deduced by using a longer rotation period, 36 hrs.  By doing this you erroneously converted a rotation period into a false period of revolution, ending up with the smaller (erroneous) semimajor axis of 7/8 AU. By way of comparison, for example, though Jupiter's day is much shorter than Earth's ( »   10 hrs.) that doesn't mean you change it to 2.4 Earth days! Its period of revolution is still measured in standard EARTH years. 

The crucial point is the value of the orbital period of your fictional planet to make one circuit is: 438/365 = 1.2 yr. =  P'

To get its semimajor axis (call it a') we use the general form for the Kepler 3rd law equation given the period  P' (1.2 yrs.): 

(P/ P' )² = k(a'/ a)³

Where P, a are the standard Earth values (period, semimajor axis) and P', a' are your own model values.

You assigned:  P' = 438 d 

= 438d/ 365.25 d/yr = 1.199  »   1.2 yrs.

Which goes into the Kepler eqn., viz.

(P/ 1.2 yr)² = k(a'/ 1 AU)³

So we now just need to find a':

Using basic algebra to solve for a':

a' = {[P']²}^1/3  

a' =  {[1.2 yr]²}^1/3  

=  {(1.44 yr)² }^1/3  

Or, on solving for a' (and remembering the units used):

a' = {1.44 }^1/3  = 1.129 AU »   1.3 AU

The '292 days' is erroneous in any case - as I showed- so is irrelevant in working out the axial tilt - which can't be computed from any  Kepler equations. Kepler's Laws and equations are applicable entirely to calculating a planet’s orbital mechanics (e.g., speed, position in space, and orbital period). 

Because axial tilt relates to the orientation of the planet’s spin axis relative to its orbital plane, it is completely independent of these laws. Calculating a planet's axial tilt requires entirely different methods, such as observing how the apparent position of the central star changes in the planet's sky over the course of a year. I.e. measuring the planet’s Sun's maximum altitude at different seasons or tracking long-term shifts in the planet's gravitational pull.

The main thing about Kepler's 3rd is always to bear in mind the units are AU and Earth Years. This never alters for any supposed solar system work, so you can't just lengthen the rotation period to obtain a different period of orbital revolution. 

 Re: axial tilt, consider from the diagram below the axial tilt of the Earth at 23 ½  degrees,  which is also the tilt of the ecliptic with respect to the celestial equator (determined from Earth)

Since you demand 11 degrees axial tilt for your planet this means the ecliptic angle with respect to its celestial equator will be 11 deg not 23 ½  degrees.  Comparing again to our Earth system and orientation - say for defining the seasons,

So for the total planet year - or 438 day period -  and 11 degrees being the maximum tilt (i.e. at its summer solstice, call it day 0) we see how it moves to its first equinox (say from day 0 through day 110), whence it then moves back to the equator (day 111-221), then 11 degrees south, and so on and so forth. Basically we have + 11 degrees at your planet's solstices, analogous to + 23.5 degrees for Earth's

 Thus all the same behavior applies as to our Earth -- except now you re-do this for the total 438 day period, and 11 degrees is the maximum moved north (say from day 0 through day 110, e..g.  A to B), whence it then moves back to the equator - to get to your autumnal equinox.  Then B to C (day 111-221), to get to  maximum 11 degrees south (for your winter solstice), and so on and so forth.

Assuming you construct a similar latitude and longitude system things will also be different.  Mainly the longitudes, given latitudes will remain the same, except 12.5 deg less tilt means more solar illumination at the summer solstices (more daylight hrs.) and less at the opposite (winter) solstices.

 Back to longitude. On Earth longitude meridians are separated by the fact it turns through 360 deg in 24 hrs, so:

360 deg/ 24 h  = 15 deg/ h   

So each longitude meridian is 15 degrees apart.  But for your planet, since you assign a 36 hr day:

360 deg/ 36  h  = 10 deg/ h   

You will end up with 36 longitude zones (time zones) of 10 degrees each compared to 24 for Earth (of 15 degrees each).

Given your designated axial tilt (11 degrees) the Sun for your planet will also assume different observed azimuth values for different dates.

Consider, on Earth we designate the Tropic of Cancer at 23.5 N. For your planet it would be 11 N.  Similarly, on Earth the Tropic of Capricorn is at 23.5 N. For your planet it would be 11 S.

Consider the azimuth for the respective suns (Earth's and yours) at summer solstice. The Sun's azimuth (A) for a given latitude (lat) on Earth for that date can be computed from:  

cos (A) = sin (23.5)/ cos (lat)

Since 23.5 degrees is Earth's axial tilt.

For your planet is would be:

cos (A) = sin (11)/ cos (lat)

Since 23.5 degrees is Earth's axial tilt.

Referencing Earth, and taking London's latitude of 51.5 N  we'd get:

cos (A) = sin (23.5)/ cos (51.5)

Or  »  49 degrees  North of due east

But for your planet, using the same latitude of 51.5 deg:

cos (A) = sin (11)/ cos (51.5)

Or A =  72.1 deg N or E. 

  Hopefully, these corrections and elaborations will help you in developing your fictional planetary model which - for good science fiction - always needs to be faithful to actual physical laws!