Chapter 6: Stellar Parameters

 Stars are originally formed from local condensations of interstellar gas and debris.
As the dust cloud contracts, it forms a relatively dense opaque sphere which is not yet hot enough for the occurrence of thermonuclear reactions.
The protostar continues to contract under its own increasing gravitation and begins to convert gravitational potential energy into heat and light. As time passes, the protostar continues to contract while its surface temperature increases. At a critical temperature, the atoms and ions within the interior of the star have begun moving fast enough to exert a balancing pressure against the weight of the overlying material. At this time, contraction ceases and a condition of near equilibrium begins.

The Main Sequence
Once the young star has reached equilibrium, it’s luminosity and surface temperature remain stable for a very long time. Stars having achieved this long-term stability are  said to be ‘Main Sequence stars’.

Please study the following, Hertzprung-Russel Diagram. As you look at the narrow sequence of stars that run across the H-R Diagram, from the upper left to the lower right, note that stellar luminosity and surface temperatures decrease incrementally between spectral class O through spectral class M.
As you move from upper left to lower right in the the diagram, the Main Sequence stars become cooler and less bright.
Its difficult to show in the diagram, but the color of the stars change as they become cooler, with the hottest stars being blueish, then white hot , then cooling to white-yellow, yellow, yellow-orange, orangish and finally, to small reddish stars.
On a clear night when you’ve been outdoorslooking at the stars, did you notice that some twinkle bluish, some white and some twinkle reddish? The color you saw is related to the star’s temperature and its stable period.

Residence on the Main Sequence
A star’s initial mass determines it’s length of stability, hence residence on the Main Sequence. See Table: Stellar Parameters below; and compare Mass (where p, our Sun, =1 for comparison) with a Stable Period (in billions of years) for the various stellar classes.

A large, hot, luminous star like a spectral class F2, uses its fuel rapidly, becoming unstable after maybe 3.47 billion years. A  smaller, cool, dimmer star, like the G9 spectral class uses its fuel much more slowly, giving it a stable period of 9.3 billion years. Other examples of this are;
•  A star of Spectral class B0 has a mass roughly 17 times that of our Sun, a surface temperature of about 20,000K and  a Luminosity 30,000  times greater than the Sun. This class of Main Sequence star radiates a very large amount of light and heat into space. Quickly using up its hydrogen fuel, it provides only an 8 million year period of stable energy output.
•  Stars of Spectral Class K0 have a Mass of only 0.74 and a Luminosity 0.28 of our Sun and a Surface Temperature of 4685K.      The K0 class of stars radiates relatively small amount of light and heat into space; slowly burning  their hydrogen fuel, they have a 28 billion year stable period of residence on the Main Sequence.

Lets assume that a star must remain stable for 3-1/2 billion years before intelligent, tool using life develops in its ecosphere. This assumption means that evolution occurred there considerably faster than it has on Earth; a process that could be accelerated by other stellar or planetary surface conditions.

A star of Spectral Class F2 has a 3.47 billion year residence on the Main Sequence, it is here that we will draw the upper limit for stellar luminosity and surface temperatures.

Table: Stellar Parameters

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Table: Tidal Retardation and Stellar Spectral Class

Spectral   Class

<   G6

G7

G8

G9

APST   restrictions –
if any.

All habitable APSTs may exist through G6 class stars. 45ºC  to 60ºC max. APST for G7 class stars. Only 45ºC and lower APST for G8  stars. 30ºC and lower for G9 class stars.
Spectral   Class K0 K1 K2   >

APST   restrictions –
if any.

Only 15ºC and lower for K0 stars. Around   0ºC APST only for K1 stars. There is tidal retardation over the entire habitable APST range.

APST= The average planetary surface temperatures used in this study, ranging from oºC to 60ºC.

Between G7 and K1, only those planets with progressively cooler APST can be considered for this study. We will draw the lower habitable limit of stellar luminosity and surface temperature at spectral class K1.

Planetary models with average surface temperatures between 0°C and 60ºC can only exist in orbit around stars of spectral class,
F2 (upper limit), due to it sshort 3.47 billion year stable period and K1, below which the planet’s rotation is stopped.

Computing Sidereal Period and Orbital Radius
Early in this study, we learned that by starting with a planet’s average planetary surface temperature (APST), we could trace our way up through the atmosphere, adding on the Albedo energy losses, etc. and finally end up with the planet’s Solar Constant  at the top of the atmosphere.
We can now choose a Main Sequence star between Spectral Class F2 and K1, and by using the planet’s Solar Constant, determine the planet’s Sidereal Period (year length) and Orbital Radius (distance from star).
Please note that the Orbital Radius, Spectral Class and related Sidereal Period, are conditions which help create the Solar Constant on a model planet.

In the next graph, Computing Sidereal Period and Orbital Radius, I have eliminated the computations so that one need only read the graph in order to determine a planet’s sidereal period and orbital radius.

Let’s read through the graph once to see how it works:
Suppose that your planetary model has a Solar Constant of 1.5 Earth’s and that you wish to place this model in orbit around a F2 Spectral Class star:
Step 1.  Locate the 1.5 Earth equivalent Solar Constant on the bottom of the graph and read up until you intersect with the F4 Spectral Class curve.

Graph: Computing Sidereal Period and Orbital Radius

Step 2. From this intersection, read directly to the graph’s left margin and read the planet’s Orbital Radius, which in this example is 1.25 AU (25% further from the F4 star than we are from the Sun).
Step 3. Using a ruler or other straight edge, read from the 1.25 AU across the chart to the far right, until you intersect the curve on the Sidereal Period graph.
Step 4. From this intersection, drop a line straight down to the bottom of the graph and read the planets sidereal period (length of year relative to Earth), which in this case is 1.4 Earth years.
.
[Note, if you have turned to the graph above, Computing Sidereal Period and Orbital Radius, from the graph, Determining Solar Constants in Chapter 2, you will need to convert the ‘actual Solar Constant’ from cal/cm2 min into the ‘solar constant in Earth equivalents’ (where Earths normal =1) , by dividing the model planet’s Solar Constant by 1.97 cal/cm2 min. Usethe equation below, otherwise use the previous steps to determine  Sidereal Period and Orbital Radius.)
Conversion factor, if needed:  (2.95 cal/cm2 min)/ (1.97 cal/ cm2 min)= 1.5 as used in the example above.   ]

If you wish to do the calculations on your own, the equations are provided below,

R = /(L/Se)  and  R3= P2

R  = Orbital radius in Astronomical   Units (AU)
L  = Luminosity of the star relative to that of the Sun, so that  Š =1
Se= The model planets Solar Constant relative to Earth, so that  Ê = 1
P  = Sidereal Period, relative to Earth, so that Ê = 1

To convert orbital Radius from AU to miles, multiply

R (miles) = R(AU) * 93,000,000 miles/AU

 

Converting 24 Hour Earth Days to Alien Planet Days
In the previous example, we found that a planet with a Solar Constant of 1.5 (relative to the Sun), orbiting  a F4 Spectral Class star has a Sidereal Period 1.4 times as long as Earths, so that,

1.4 (sidereal period) * 365 days per Earth year = 511 Earth days per year for this particular model planet

During our discussion of planetary rotation in Chapter 2, we found that a planet’s mass is related to it’s rate of Rotation. See Table: Planetary Parameters, row ‘Rate of Rotation’, Chapter 2.  Note that a planet of 2.0 Earth mass has a 14.4 hour rate of rotation; it’s day is 9.6 hours shorter than our  24 hour day length on Earth.

The preceding equation gives the planet’s year length (Sidereal Period) in terms of Earth days. It is important to convert from Earth length days to alien planet length days, in order to gain a clearer perspective of the interaction between life forms and their duration of exposure to heat, cold, light, ultraviolet radiation, etc.

Consider the following examples;
•  If an American service worker spends 1-1/2 hours per day commuting to and from work, has a 1 hour lunch and 2 each 15      minute work breaks, he has spent 3 hours or 12.5% of an Earth standard 24 hour day, basically idle. If an alien worker living  on a 2.0 Earth mass planet with a 14.4 hour day, spent as much time commuting, eating lunch and on work break, those 3 hours we took would account for 20.6% of his ‘day’.
•  On a hot, arid desert, temperatures climb rapidly during the first hour after sunrise and begin to decline an hour before sunset.
On Earth, a desert inhabitant would have about a 10 hour exposure to extreme heat; however, a desert inhabitant on our model 2.0 Earth mass  planet, with its faster rate of rotation, would only have about a 5-1/4 hour exposure to extreme heat.
•  Survival, in a habitat  hostile to unadapted life forms, depends on the degree and length of exposure to the life limiting elements.

Converting from the number of ‘Earth Days per year’ to the number of ‘Alien Planet Days per ‘year’

Equation: Pa = (Pe*365 days*24 hours)/N

Pa = Sidereal Period in Alien days
Pe = Sidereal Period relative to Earth (where  Ê = 1)
N = Rate of rotation for the model ‘alien’ planet,  in hours.

From our previous example:
How many Alien days are there on a 2.0 Earth mass model planet which is orbiting a F4 star and having a 1.4 (w=1) Sidereal Period? Using the previous equation:

Pa = (1.4*365 days*24 hours)/14.5 hours

Pa = 845 Alien days is the length of  this model planet’s year
This model planet has a year length of 845 days, while each day is 14.5 hours long.
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Stellar Rotation
It’s currently felt that stars which are accompanied by a planetary system, have transferred most of their angular momentum to the planets.
In our own solar system, only 0.1% of the mass, but 98% of the angular momentum is tied up in the planets.
If the planets did not exist, the Sun would rotate once on its axis every 12 hours, instead of once every 25 days as it does.

When considering Main Sequence stars, there is a large decrease in the rotation rates between spectral classes O through spectral class A; however, the rate of rotation remains fairly constant between spectral classes F through S, this is thought to be the result of these cooler stars having transmitted their angular momentum to planetary systems in their domain.

 Table of Observed Stellar Rotation Rates

Planetary Systems
The space environment, around potentially habitable stars systems spectral class F – S, is likely to contain many of the same planetary orbiting bodies as are found in our Solar System. The general mass range of these bodies may include:  one or two stars, super planets, rocky planets, planetary satellites (moons), asteroids and comets; our system contains all these bodies except a super planet.

Alien intelligence may find their solar system containing a different numerical mix of orbiting masses, but if they ever develop a technical phase they would deduce the same laws regarding nature as we have.

Whether you have a 24 hour day and 365 day year, a G2 class star, 1.0 Earth mass planet with cool temperatures, or a 14.5 hour day and 845 day year, F4class  star, 2.0 Earth mass planet with warm temperatures is immaterial. The laws of physics and chemistry are the same on both and all planets.
Water will freeze or evaporate under similar conditions of temperature and pressure. Carbon will form with oxygen to make carbon dioxide, rain will soak into the soil and excess will run off in rivulets seeking to accumulate in lower areas, ie lakes or seas.
The poles on planet’s with low axial inclination will be much cooler than the equatorial regions.
Life will develop to make use of and extract stored energy in the environment.
During much of the planets habitable period there will be a Darwinian explosion in the number of life forms, representing greater diversity with occassional environmentally imposed fluctuations. Later, as the parent star becomes less stable, there will follow a several million year Post Darwinian collapse, extant life will become simpler in a habitat less capable of supporting life.

Continued in Chapter 7: Morphology of Intelligent Life

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Filed under SRAPO (Stellar Radii and Planetary Orbits), __Chapter 6: Stellar Parameters

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