Forgive (Yourself) and Forget (Everyone Else)

Sometimes I feel like one of those wind-up toys. On particularly bad days, stress and tension build up until I'm at my limit and I just have to go bonkers for a bit before I can continue on with my life. Being naturally curious, I try to identify when and how these nuggets of negative energy accumulate and at what point the critical threshold is reached before I can't handle any more.

In this sense, a better analogy would be that of a balloon. In this example, all of those negative thoughts and feelings that we tend to hold on to would be the air inside the balloon, and the emotional limit would be the point at which it pops. The size of an individual's balloon can vary, along with the rate at which air enters and exits.

For those of us who happen to be very self critical, we tend to internalize this negative air quite frequently and let it create tension within our mental and emotional balloon. In addition, such an attitude often makes it difficult to let go of perceived troubles and release pressure. The result is the presence of a lot of unnecessary stress and anxiety that can only hinder our ability to live a more fulfilling life. After all, it's quite difficult to stay grounded when you're filled with so much hot air (I'm having too much fun with this).  Eventually, the balloon has to pop, and all of the pent up energy is released in the form of outbursts, breakdowns, etc.

As someone with a busy mind, I occasionally make the mistake of interrupting people in an effort to remain engaged in conversations. Sometimes it feels like I have an unfortunate sixth sense for when someone's about to start speaking, because on many occasions, I can enter a conversation at the exact same time as someone else. Whereas a healthy response in this situation would be to apologize, let go, and move on (I'm getting better at that), I instead fixate on the awkward and grow very self conscious of when I choose to speak up. Rather than actually focusing on what others around me are saying, I retreat into my own head in order to manage the building stress from all the negative thoughts I'm holding on to.  As expected, this tends to exacerbate the situation.

If there's one thing I learned from the internet, it's that you shouldn't read the comments section on most websites.  Similarly, it's probably best not to read the comments section of your mind.  So what's the moral of the story? We're all human and we all slip up sometimes, so let yourself feel embarrassed, forgive yourself, forget about what others are thinking, and continue being awesome.

Planets in Binary Systems

Lucasfilm

In the previous post, I discussed how the oscillating orbits of planets within binary star systems could create extreme climate cycles as a result of the fluctuating energy input.  What exactly is going on in these systems, though? In this post I seek to describe more technically what is referred to as the Kozai-Lidov mechanism by which a second star can alter the shape of an inner planet's orbit.

In binary systems, planets can orbit in two configurations. The first, like that on Tatooine in Star Wars, has the two stars at the center with the planet orbiting around both. For these circumbinary systems, dynamical stability requires that the planet orbit at a distance significantly greater than the separation between the two stars.

The second configuration, which my research focuses on, has the planet orbiting one of the two stars, with the second circling the system further out. Under these circumstances, the gravitational influence of this stellar companion can augment the eccentricity of the planet's orbit  (science speak for making the orbit more elliptical) via the Kozai-Lidov mechanism.  For this to occur, the orbits of the secondary star and the planet must be inclined relative to each other by a sufficiently large angle (approximately 40 degrees or greater).

In the setup above, the planet (p) orbits the primary star (1). The secondary star (2) orbits with a large inclination angle i relative to the orbit of the planet.  The angular momentum of the planet in the direction parallel to that of Star 2 is conserved.

Under this configuration, the angular momentum of the planet's motion in the direction parallel to that of the secondary star is conserved. This quantity, denoted as Lz, depends on the eccentricity of the planet's orbit e along with the inclination i.  As Star 2 tugs on the planet, the decreasing inclination between the two orbits is traded for eccentricity, and the planet's orbit becomes more elliptical.  This dynamical exchange between eccentricity and inclination occurs in cycles with a frequency that depends on the masses of the stars and the radii of the two orbits.  The oscillation occurs more rapidly for a system with a more massive secondary star, a less massive primary star, and a planet which orbits further out.  This means that the effect is most prominent when the secondary star has a substantial gravitational influence relative to the primary.  The maximum eccentricity that the planet can reach depends only on the initial inclination angle.  If the orbit of the secondary star is eccentric, however, Lz is not strictly conserved and arbitrarily high eccentricities can be attained.  The phenomenon was originally studied with reference to asteroids in highly elliptical orbits that were influenced by Jupiter's gravity, but in recent years has been applied to the dynamics of extrasolar planets.  

These cycles can be visualized in the plot below.  An eccentricity of zero corresponds to a circular orbit, while an eccentricity of 1 is the maximum limit.  Higher eccentricity corresponds to a more elliptical orbit. The setup involves a planet orbiting a star 1.4 times as massive as our Sun at a distance twice that of Earth. The secondary star is 0.4 times as massive as the Sun and orbits at a distance 20 times greater than that of Earth's orbital radius.  Under this configuration, the planet's eccentricity cycles from nearly zero to approximately 0.4 over a timescale on the order of 10,000 years. At this eccentricity, the average stellar flux received by the planet would be about 1.1 times greater than if the orbit were circular. If Earth's orbit were this elliptical, its annually averaged temperature would be warmer by about 5 degrees Celsius, although seasonal temperature variation would be substantially greater.  Within one orbit, the planet's energy input would vary by a factor of over five between the nearest and farthest points from the primary star.

So what are the implications for planetary habitability?  One potential research question would be whether or not these cycles could thaw a planet like Earth out of a state of global glaciation. On the other end, these spikes in eccentricity may render an otherwise habitable planet too warm to sustain liquid water.  Under these circumstances, would the planet retain its total water supply in the atmosphere, or would it lose vapor to space and dry out over the course of its lifetime? Such questions pertaining to the habitability of these planets have yet to be investigated.

Planetary Habitability

Within the past decade, we've discovered a lot of new planets around stars other than our Sun (for an up-to-date catalogue, visit www.exoplanets.org)  These extrasolar planets, or exoplanets for short, can exist under conditions very different than for planets in our own Solar system. From Jupiter-sized giants orbiting their host star more tightly than Mercury to tidally locked rocky planets with permanent day and night hemispheres, the diversity of observed worlds is impressive and suggests that habitable worlds may differ substantially from Earth.

The most broadly accepted definition for a 'habitable' planet is one which can sustain liquid water at its surface. While it is theoretically possible that life could arise by other means, life as we know it depends on the presence of liquid water. In our search for exoplanets that could sustain life, we are mainly interested in identifying planets within a range of orbital distances referred to as the 'habitable zone.'  

While the stellar flux (ie the radiative energy received by a given surface area over a certain amount of time, a quantity related to temperature) depends only on the orbit's radius, surface temperatures (and by extension the habitability of the planet) may be governed by the climate system.  Venus, despite being at the inner edge of our habitable zone, has a thick carbon dioxide atmosphere which leaves the surface too warm for life to develop. At the other extreme, planets with no atmosphere cannot be habitable, as liquid water cannot exist at zero pressure.  

Over a planet's lifetime, the stellar flux may vary due to several processes: [1] Stars getting brighter with age [2] Planets migrating inward or outward due to dynamical interactions [3] Orbits may not always be circular. With a changing energy input, the habitability of a planet over time depends on the stability of the climate system and its feedback processes. Positive feedback processes amplify temperature change.  One example is related to the reflectivity of ice vs oceans. Increasing global temperatures would melt ice, thereby reducing the planet's overall reflectivity and allowing for more absorption of light.  This would increase temperatures even further.  One primary negative feedback which maintains equilibrium here on Earth is related to weathering processes between the atmosphere's carbon dioxide and minerals at the surface. A positive perturbation to global temperatures would melt ice, thereby exposing more land and increasing average precipitation rates. As a result, weathering rates would increase, thereby pulling more carbon dioxide out of the atmosphere, weakening our greenhouse effect, and reducing temperatures back to equilibrium.  The strength of these feedback mechanisms can influence a planet's resilience to changes in thermal forcing.

On Earth, our orbit is nearly circular, allowing for a more stable climate system over time. This may not be the case, however, for many planets within binary star systems. With the presence of a second star beyond the orbital radius of the planet, gravitational dynamics may warp the shape of the orbit to be more elliptical, thereby varying energy input over the course of the orbit.

At sufficiently high angles between the orbit of the planet and the secondary star, the shape of the planet's orbit would oscillate. Over short enough timescales, these oscillations may create potentially extreme  climate cycles. Depending on the strength of the climate's feedbacks and how elliptical the orbit becomes, a planet like Earth could periodically freeze or boil over if it were to temporarily exit the habitable zone. Whether or not such events would render the planet permanently uninhabitable would have to be determined via climate modeling.

With the gravitational influence of a second star (upper right), the shape of a planet's orbit can oscillate between circular and elliptical, causing potentially extreme climate cycles. Note: The timescale over which the orbit changes shape is much longer than that over which the planet orbits the primary star.

Given the growing number of planets detected within binary systems, it is important to understand the limits of habitability under the physical conditions that would be expected with the presence of a second star.  If such worlds could indeed maintain stable climates with liquid water, the extreme climate cycles would present developing life with many challenges that were never present on Earth. I'll leave those topics to the biologists.

I'll end this post with some imaginative thinking about what a double sunset would look like on such a planet. Perhaps two stars may be better than one.

Public Domain Image Modified by Nathan Baskin

Random Musing of the Day

"Frantically seeking closure is a futile endeavor when life throws you open-ended questions."