Download Planet Notes 2 - 1 Notes 2: Physics of the solar system 2.1 Basics of

Notes 2: Physics of the solar system
2.1 Basics of Orbits
It is time for you to remember all of the rules of physics that influence planets. And we’ll start
off with the basic motion of solar system bodies which are defined by Kepler’s Laws of
Planetary Motion. But remember, they apply to anything orbiting anything else, not just planets
orbiting the Sun.
And as you should recall there are three laws. The first defines the orbit as an ellipse, with the
Sun found at the focus (pl. foci). With the old model of the solar system (that of Ptolomy) the
orbits were all assumed to be nice and neat circles. That is not the case – orbits can be circles,
but all known orbits are elliptical and that makes it a little more complex to define the locations
of objects in their orbits. For an eclipse you have the following parameters –
a = semi-major axis (so the size along the longest or major axis = 2a) . This also equals the
average distance between the Sun and a planet.
b = semi-minor axis
(perpendicular to the semimajor axis)
e = eccentricity of the
ellipse, which defines the
degree that it is removed
from a circle. Values for e
range from 0 to <1 for
ellipses.
The value for e can be found
by taking
e = (distance between foci)/major axis.
The value for e has no units of measure – it is dimensionless.
With these locations defined it is possible to determine at any given moment the distance
between the Sun and an object in orbit about it.
ro 
a(1  e 2 )
1  e cos
2-1
where ro = distance between the planet and the Sun and  is
the angle around from the location of perihelion (the
closest location shown). In this formula ro and a have the
same units of measure which should be in meters, especially if the values are needed in other
formula.
Planet Notes 2 - 1
The speed at which a planet travels around the Sun varies with distance from the Sun, and that’s
the basic upshot of Kepler’s second law. A relationship for the velocity of an object in an
elliptical orbit is given by
 2 1
2-2
v 2  GM   
r
a
 o

The units here should be in kg, meters, seconds, etc. As ro gets smaller (closer to the Sun), the
value for the first term gets larger and the resulting velocity is larger – near the Sun objects move
faster! And as ro gets larger, the first term gets smaller and the resulting velocity is smaller –
objects far from the Sun move slower! Wow, that’s exactly what Kepler’s second law says. So
it is pretty easy to see how fast objects are moving so long as you know where they are located in
their orbit.
And of course Kepler’s third law is the famous relationship
2-3a
P 2  Ka 3
Where P=period of the orbit about the Sun, a=semi-major axis/average distance from the Sun,
and K=constant. For the special case of an object going around the Sun and when measuring the
period in years and the average distance in A.U. (Astronomical Units), the value of K=1. And
the relation reduces to
2-3b
P2  a3
Later Newton was able to provide a reason for the orbits that were based upon the Law of
Gravity, and was able to refine the relationship to be
P2 
4 2 a 3
G(M 1  M 2 )
2-3c
where M1 and M2 are the two objects (the orbiter and the orbitee). With this version it can be
expanded to any orbital situation, not just planets around the Sun. Usually the mass of one
object is much larger than the other mass, in which case you can usually ignore the smaller mass
in the calculations. In that case the formula becomes
4 2 a 3
P2 
2-3d
G ( M big )
This is a very helpful formula to use to determine the mass of a large object, since you only need
to watch the small mass object’s orbit to figure out the mass.
In order to define an orbit in the solar system precisely, several parameters are needed such as
a = semi-major axis
e = eccentricity
i = inclination (tilt)
q = perihelion distance
 = longitude of the ascending node (angle around from vernal equinox)
 = argument of the perihelion (angle from node to perihelion location)
Planet Notes 2 - 2
The last two define locations of
the perihelion position of the
object relative to a point, usually
the Vernal equinox position.
Other values are needed for
positions of comets and
asteroids, such as the date of
perihelion and the epoch for the
coordinates (when were they
defined).
2.2 Other Types of Orbits
While planetary orbits are pretty
simple, you have to remember
that some objects go into and out of the solar system with non-elliptical or non-circular orbits.
The velocity of the object can often determine whether the object is trapped into a stable orbit
around something (like the Sun) or if it will be able to escape from the Sun and only pass
through the solar system one time.
So that means you need to know what the escape velocity of an object is to determine its fate.
The escape velocity is given by the formula
2GM
vesc 
2-4
r
Where M is the mass that you are trying to escape from – the Sun for most solar system objects.
Of course the amount of velocity you need also depends upon how far away you are from the
object (r), but apart from the mass and distance, not much else matters. For objects in circular or
elliptical orbits, those always have velocities< the escape velocity.
In cases where the velocity of the orbit is equal to the escape velocity of the system, then the
object will be in a parabolic orbit. This is actually pretty rare, and would be a precise balance
between the energy of motion and the gravitational energy. In that case if you want to define the
position of the object, like was done previously for a Keplerian motion, you’d have
2q
2-5
ro 
1  cos
Where q = closest approach in meters, like ro , and e=1 (it is actually in the formula, but doesn’t
really contribute).
And the next case, that where the velocity > escape velocity, is the case in which you’d have a
hyperbolic orbit. This results in the relationship for the location defined by
q(1  e)
2-6
ro 
1  e cos
And you also have the aspect that e>1 for hyperbolic orbits. This is pretty common situation
with the orbits of comets.
Planet Notes 2 - 3
2.3 Three-Body Problems
So far we’ve just looked at how two objects move, usually one object going about the Sun.
Since there are quite a large number of objects in the solar system, that is a rather simplistic way
of looking at how things are set up. However, once you get beyond predicting the motion of 2
objects the math goes crazy. Why is that?
If you just look at the basic gravity formula you have only two masses involved there. But if you
have 3 objects (called 1, 2 and 3), then you have to continually calculate the influence of all three
objects on one another. So how does 1 impact 2, and how does 2 effect 3 and how does 3 react
to 1? And all of these interactions have to be calculated instantaneously, since the motion or
change in one object impacts the other two, which also impact each other and the first one, and
…well it gets really annoying after a while trying to keep up with it.
Since it isn’t practical to make such calculations it is best to use some simplifications or
approximations to calculate the complex motions of the system. One way to do this would be to
take small steps in predicting motions, such as where should the objects move based upon their
current motions, where should they be, how likely will they effect the other objects around them,
and so on.
This is the basic set up for a 3-body problem, and in spite of the name, it is not as easy as 1, 2, 3.
In some cases small scale changes like perturbations, can be quite easy to work with. Often
perturbations can be long term and predictable (periodic), so changes are easy to calculate.
There are also the situations where objects cycle around and get back to similar orientations that
are repeated. This would be a commensurable orientation (or orbit). This is often seen in
objects that have orbits that are harmonics of one another. For example if the orbital period of
one object is 1 year and that of another object is 2 years, then they’ll be back in the same location
relative to the Sun and each other every two years. So basically look to see if the orbital periods
of the objects involved make a simple fraction, like ½, 1/3, ¼, 2/5, 1/5, ¾, etc. This sort of
periodicity causes a repetition of alignments and increases the influence of these objects on one
another. And if one of the objects in this predictable re-alignment is relatively large, then rather
interesting things can happen. These situations give rise to resonances – a “beat” phenomena of
orbits that cause disturbances in the distribution or motion of objects. Resonance locations are
found in the asteroid belt (the Kirkwood Gap) and amongst planetary rings (Cassini Division).
Another situation for 3 bodies is brought up in the Lagrangian Points. These are gravitational
“peaks” and “valleys” associated with two objects. There are 5 Lagrangian points (locations)
that are defined around 2 objects, usually with the designation of L1, L2, and so on.
L1, L2, L3 are all unstable (the “peaks”)
L4, L5 are stable (to a degree)
Planet Notes 2 - 4
What does that actually mean? Generally
speaking if an object is at L1, L2 or L3, it
has to stay precisely at that location for it
to remain there. If it bumped even just a
little bit, it will fall out of position.
Objects at L4 and L5 tend to stay in that
area even if they are bumped. But if they
are really pushed then they can leave that
location. It is worth noting the relative
locations/distances of the objects to the
Sun. L3, L4, and L5 are all the same
distance from the Sun as the Earth, so they
orbit around the Sun in the same amount of
time that the Earth does. L1 and L2 are at
slightly different distances, but the
gravitational influence of the Sun and
Earth defines that location.
In our solar system there are quite a few things at the Lagrangian points, particularly L4 and L5.
Examples include The Trojan asteroids, most at the Jupiter-Sun L4, L5 locations – there are about 5735 of
those
Excess dust found at the Earth-Moon L4, L5 locations
Satellites are often put in the L1 and L2 locations for the Earth-Sun system
Soho is at L1, and in the future LISA
WMAP, Herschel and Planck are at L2 and in the future the JWST
(These satellites have to regularly correct their positions to stay in these locations)
For L4, L5 Earth-Sun system we find mainly dust, and one Trojan
For Mars-Sun, find about 3 Trojan asteroids at L4, L5 locations
For Saturn-Dione, you find two other satellites, Helene, and Polydeuces at L4, L5
For Saturn-Tethys, you find two other satellites, Calypso and Telesto at L4, L5
For Neptune-Sun find about 9 Trojan asteroids at L4, L5
There is a continual effort to find other Lagrangian point objects around other objects, like Venus
and Uranus, but at this point none have been found. Also it is rather interesting that there aren’t
any objects in the L4, L5 locations for Saturn (that we know of).
There are a couple of other interesting ways of looking at how objects move, though it is all a
matter of perspective. Horseshoe and tadpole orbits are observed when we view the motion of
an object while keeping other objects fixed. Technically nothing moves in an orbit that has a
horseshoe shape, but when you fix the Earth for example and see how other objects move
relative to it, these objects trace out a horseshoe shape.
For example, you are used to seeing the following diagram for the motion of the Moon about the
Earth (on the left). Obviously the Earth isn’t in this one position all the time, so the motion that
is actually seen is more complex, more like the picture on the right (yes, poorly drawn).
Planet Notes 2 - 5
One of the most famous examples of a horseshoe orbit is seen in the moons of Saturn, Janus and
Epimetheus. These objects move in a way such that as you view the motion from one, it looks
like the other traces out a horseshoe. So for a while (from the perspective of Epimetheus) it
looks like Janus first comes towards it, and then shifts into an inner orbit and moves away from
it. What is really happening is that the object located closer to Saturn moves faster in its orbit, so
when it catches up to the slower outer orbit object, it would appear to be coming towards it.
Then the two objects swap locations and the faster inner object is now the slower outer object, so
it looks like it is going away (when in fact the inner object is just moving faster). It takes a while
for this to happen for Janus and Epimetheus, about once every 4 years the moons switch places.
The last switch was in 2010.
Generally speaking a horseshoe orbit is seen only when you have objects that are switching from
faster to slower (inner to outer) orbits. This can also happen when one of the objects is not as
large as the other, and it is the only one to switch its orbit. In the case of Epimetheus and Janus,
the two objects are similar in size so they both move. There is actually a rather nifty situation
where the Earth has some horseshoe orbiting objects that come close to it.
Cruithne is an object with a near resonant orbit with
the Earth (1:1), so that the time that it takes to go
around the Sun is nearly equal to the time it takes the
Earth to do the same thing. But since Cruithne’s orbit
isn’t exactly in a 1:1 relation with the Earth’s orbital
period, its orbit “moves” around the Earth in a
horseshoe shape. In the short term the motion of
Cruithne’s position from our non-moving perspective
is like a kidney shape. But the kidney shaped orbit
slowly moves, such that it takes 770 years for one
“orbit” of the kidney-orbit. The object gets close to us
every 385 years which corresponds to the ends of the
Planet Notes 2 - 6
horseshoes. The next time this happens will be in July of 2292. At that time Cruithne will be
only 12.5 million km away from the Earth. What is happening in this situation is that the Earth
is perturbing the orbit in such a way so that it slowly migrates to an alternately slower or faster
pace about the Sun and it ends up moving towards us slowly or away from us. While Cruithne is
not really in orbit about us, it does interact with us, and some people have gotten to calling it
another Earth satellite. While we do influence it, there is a large amount of time that the thing is
nowhere near us, so it should not be viewed as a satellite.
Another object with a funny orbit is 2002 AA29, which has a really catchy name. This one is in a
spirally horseshoe orbit and it will gets close to the Earth every 95 years. In about 600 years, the
object will be for a while another satellite of the Earth (quasi-satellite), since the loops put it
around the Earth for a while, and at that time there will be two satellites around the Earth. At
least for a while. Technically it isn’t a real satellite since it doesn’t stay in that location, but is
just moving along. It is likely that this situation has happened previously, in the 6th century BC,
but it is not a very large object so odds are no one could have seen it in the sky. Each loop of the
orbit takes about 1 year.
There are also some other objects such as Izhdubar, 2000 PH5, 2003 YN107, 2004 GU9, and 1998
UP1 that are also close to a 1:1 resonance with the Earth and do sort of act like satellites, but
usually not for very long. So it doesn’t mean that we have a half a dozen or more moons, just 1
moon and a bunch of other objects that meet up with us every once in a while. Actually other
planets also have quasi-satellites which may appear to be associated with a planet, but are either
temporary or just in a coincidental orbital resonance.
The other unusual orbital shape, again when one object is held fixed, are tad pole orbits. These
are really just the wobbling/perturbation of objects in the area of the L4, L5 locations. It could
be best described as a marble that gets knocked up and down in a bowl. It would take quite a bit
of a gravitational boost to knock something out of the L4 and L5 locations, so objects in these
orbits can be quite wobbly over time.
2.4 Hill Spheres
With all of the above discussion of objects getting perturbed by the Earth and other worlds, you
may wonder how does an object exert its influence upon another, and how far does that influence
go? As is usually the case it is really a matter of mass and distance (the two most important
aspects of determining gravitational influence). But technically the biggest thing is the Sun so
doesn’t that really just control everything? No. If objects are far from the Sun, they can fall
under the domination of other objects. For example, our Moon has an orbit that is defined by the
Earth. So it is under our influence. But how far away from the Earth would you have to be for
something to no longer fall under our gravitational influence? That is where the Hill Sphere
comes in.
The Hill sphere defines the limits of an object’s influence and is given by


m2

RH  a(1  e)
 3(m1  m2 ) 
1
3
2-7
Planet Notes 2 - 7
Where m1, m2 = are the mass of the main object and the smaller object that is trying to control
stuff (so m1= Sun, and m2= planet usually), a=semimajor axis and e=eccentricity. So for a
planet to hold onto a satellite, the satellite must be located within the planet’s Hill Sphere. This
would mean that the planet “controls” the satellite, not the Sun. Of course for some planets it is
possible that they have only temporarily captured other objects, though that depending upon the
object’s velocity. It is also possible for there to be a hierarchy of Hill spheres. The planets are
within the Sun’s Hill sphere, while the planets have their own Hill spheres which allow them to
have satellites, and those satellites also have Hill spheres that allow them to retain their own
satellites, and those satellites have Hill spheres which define their own regions of domination,…
and so on.
So to actually determine the Earth’s Hill sphere, you would use the Earth’s mass as m2 and the
Sun’s mass would be m1 and the distance between the Earth and Sun would be a (1.50 x 1011 m).
The value for e=0.0167. So this gives
1
1

 3

 3
m2
6.0 1024
  (1.5 1011 )(1  .0167)
  1.47 109 m
RH  a(1  e)
30
24 
 3(2 10  6.0 10 ) 
 3(m1  m2 ) 
So the Earth would control the motion of objects that are within about 1.5 billion meters
distance. This is about 0.01 AU which isn’t that great, but the Moon is found easily within this
distance.
2.5 Tides, Roche Limits and Rings
Tides are not just a measure of the gravitational pull of something but the difference in the
amount of pull an object experiences. Typically tidal forces are defined as being caused by a
difference in gravitational pull, not just gravity.
Tidal force = size across object/distance3
Obviously, the closer an object is, the stronger the tidal forces. It is also noteworthy that the
effect drops off quickly with distance. Tidal forces are in operation all over the place, most
notably in the Earth-Moon system where the tidal interactions cause several things including
 High tides and low tides of the ocean along the coasts
 A slowing down of the Earth’s rotation
 The locking of the Moon to a 1:1 orbit/rotation system, so that it has a
synchronous rotation
 The continued motion of the Moon away from us
While the Moon has the strongest effect due to its proximity, the Sun also strongly effects us –
about 20% of the Earth’s slow down is due to the Sun.
Most satellites in the solar system are similarly affected, except most don’t slow down their
parent bodies significantly. This is because most satellites are much, much smaller than the
planet that they orbit.
When can the tidal force cause damage? When things get too close! But how close is too close?
Planet Notes 2 - 8
This is how we define the Roche limit. Basically the Roche limit is how close small objects can
get to larger objects before they get torn apart. It is not a precise relation since it depends upon
the ability of the object to resist being torn apart or the tensile strength of the object that is
getting ripped apart.
In general Roche limits are given as
1
  3
RRoche   M  R
2-8a

m


So it depends upon the densities of the objects involved where M is the density of the largest
object, and R is its radius, and m is the density of the smaller object (the one that could get torn
apart).
As previously states, since compositions vary, there is no precise formula for this, but a good
approximation is
1
  3
RRoche  2.5 M  R
2-8b
 m 
This should be appropriate for most situations, such as for the rings of the outer planets.
However it doesn’t work well of unusual rings such as Saturn’s E ring.
Tides also have the tendency to not just rip things apart but to pull them so that they are
internally heated up. This is seen most notably in the case of the Galilean satellites, as well as
Enceladus and Triton.
But going back to rings, it is also important to realize that the rings are influenced solely by the
forces exerted within the Roche limit. There are also resonances that can be explained by
interactions of the rings with satellites and the influence of shepherd satellites on the distribution
of ring particles. In recent images from the Cassini spacecraft ring particles were shown to be
perturbed in a wave manner by the passage of a nearby satellite. Such motions will continually
alter the distribution and motion of ring particles, and with many satellites tugging on ring
particles, such motions are very complex.
2.6 Radiation – Light
Generally most motions are not influenced strongly by light, but small objects can be greatly
influenced by light, particularly dust or debris blown off of a comet. However larger objects are
also influenced by light as you’ll see. We’ll go through the various ways that light influences
objects from the smallest objects and we’ll work our way up to larger ones.
For submicrometer sized particles (<1 micron, or about 10-7 meters in size) Corpuscular drag
plays a significant role. This is how the solar wind influences the motion of things. Okay, so the
solar wind isn’t light, but this is really the only place that I can talk about it. The solar wind will
cause a slow down of the smallest particles due to interactions with the solar wind particles
(protons, electons, etc) with small dust sized particles. Remember, the solar wind refers to the
particles that are regularly being blown off from the Sun. Radiation from the Sun also influences
Planet Notes 2 - 9
objects (see below), but for really small particles, the solar wind has more influence than the
radiation. The end result is that the small dust particles are slowed down in their motion and
they slowly spiral in towards the Sun.
Corpuscular drag can be viewed as a process that is similar to how gas drag can alter the motions
of objects. This is most often seen in the outer layers of a planet’s atmosphere changing the
motion of objects that get into it, like ring particles, or small satellites. The Hubble Telescope is
being continually dragged/slowed by its motion in the outer part of the Earth’s atmosphere. It
would fall to the surface of the Earth if its position isn’t continually corrected.
For micrometer sized things (sizes of about 1 micron, 10-6 m) radiation pressure plays a
significant role. This is due to the fact that light or photons are packets of energy, and they can
impart their momentum on other objects. Enough light and you can alter the motion of objects.
So how much are objects influenced by light? As you would expect it depends on the size of the
object.
( flux)a 2Q La 2Q
2-9
Fradiation 

c
4r 2 c
Where the radiation force (Fradiation) depends upon the luminosity (L, measured in Joules/s), a is
the particle size, r is the distance from the Sun, and Q is the “correction factor”. This last bit is
dependent upon the ability of the object to absorb the energy, which depends upon the
composition, and the type of light (energy) that interacts with it. The value of Q varies from 0.1
to 1.0.
For centimeter sized objects, there is the influence of the Poynting-Robertson Drag. This one is
a bit more complex. First, imagine a rapidly spinning particle in space. It absorbs sunlight when
a side is facing the Sun and as it spins around, the side that was previously facing the Sun will be
able to give off (radiate) the energy it absorbed. But since the particle is also moving, it doesn’t
emit the light evenly as seen by an outside observer. There is a preference for the particle to emit
light in the forward direction, since it is moving in that way. This is due to a “relativity” type of
effect, since the energy that is emitted is skewed (not evenly distributed) in a particular direction
and this skewing alters the motion of the particle. It will slowly lose the energy of its orbit, and
tend to spiral slowly towards the radiation source. The time scale for this sort of particle motion
is on the order of thousands of years to move into the Sun from the Earth’s distance.
We see such particles influenced by the Poynting Robertson Drag in the zodiacal light. This is a
band of light visible usually around the time of the equinox when the ecliptic is nearly
perpendicular to the horizon. The particles that make up the zodiacal light band is being slowly
dragged to the Sun and are being destroyed
For meter sized and larger sized objects, there is the Yarkovsky (Yarkovski) effect, which is
another one of those energy-emission-momentum things. Again, an object is rotating so for a
time one side is facing the Sun. Eventually that side will turn away from the Sun and radiate
away its energy into space during the “afternoon”. As the object gives off light, it is also giving
off energy, which can be translated into momentum and that will alter their motion. If the object
has a prograde revolution and rotation, then the energy is given off on the back, and this will
cause the object to move to a greater orbit (since it is getting an energy boost from the emission).
Planet Notes 2 - 10
But if either the rotation or revolution is retrograde, the object will emit energy in a direction
against its motion, so it slows down and the orbit decays. So it all depends upon how you spin
and orbit.
In a similar way there is the Yarkovsky-O'Keefe-Radzievskii-Paddack effect, or YORP effect for
short, which shows that the shape of the object can alter its motion. If radiation is emitted at a
non perpendicular angle from the surface, like from a protuberance, this can change the motion
as well. Sort of like a pin-wheel rocket effect. This can change the object’s obliquity and
rotation over the long term.
Now a little bit about radiation – light – and all of that stuff.
You have to remember that there are many types of light (radio – gamma-ray), and that they
vary according to wavelength or frequency. And the most important rule is that the speed of
light is a constant (c=speed of light=constant).
Another important energy rule is Wien’s displacement law which describes the relationship
between the dominant type of light an object emits and the temperature of the object. This is
really only for perfect energy emitters/absorbers, but since things in real life can act sort of like
this, we can use the relationship
max  0.0029/ T
2-10
Where T is the temperature in Kelvin, and the wavelength is in meters.
Another useful relationship is the Doppler effect, which is particularly useful for determining the
velocity of objects. Generally this can be used to not only measure motion in space but also
things like rotation.
v observed  lab
2-11

c
lab
Where the wavelengths observed are compared to the normal wavelengths with no motion (the
lab wavelengths). Motions towards an observer result in negative velocities, and therefore
shorter wavelengths observed (blueshifted light), while motions away are positive velocities and
longer (redshifted) wavelength light.
To effectively analyze light from a planet, it is best to look at its spectrum. A spectrum can be
influenced by a variety of sources, such as the composition of the material that is reflecting the
light (material can absorb light), and atmospheres can also influence spectra by absorbing light
as well. This is one way that we can get a clue about compositions of both surfaces and
atmospheres without direct sampling. And in some cases, specific wavelengths of light are used
since they may interact better with certain types of material than just visible light.
Of course some planets not only reflect light from the Sun but also contribute some of their own.
They have an excess of energy given off due to special conditions, such as emission of energy
due to an internal sources (this is seen in Jupiter, Saturn and Neptune), or due to an abnormal
energy flow (like the green house effect on Venus). So these objects have more energy given off
than they absorb – they aren’t exactly in equilibrium.
Planet Notes 2 - 11
Another important feature of a planet is its albedo. This is basically a measure of an objects
ability to reflect light. The Bond albedo value is an average value over all wavelengths, since
different material may absorb visible light better than infrared, or UV or and radio waves better.
The Bond albedo is also averaged over all directions that it can reflect the light into, so it pretty
much covers everything with reflection. The Bond albedo varies between 0 (all absorbed, none
reflected) to 1 (all reflected). Therefore, dark objects would have a low albedo while bright
shiny objects have a high albedo. Pretty easy.
Planet Notes 2 - 12