Download Chapter 5. Determining Masses of Astronomical Objects

Chapter 5. Determining Masses of Astronomical Objects
One of the most fundamental and important properties of an object is its mass. On
Earth we can easily “weigh” objects, essentially measuring how much force the Earth is
exerting on them, which depends on their mass. For astronomical objects we can only watch
their motions in response to the gravitational interactions. A common event is when one
object orbits another (or, more precisely, they orbit each other). This occurs for planets
orbiting a star or two stars orbiting each other. It also occurs for binary galaxies although
the details of how the mass can be determined in this case are different because the orbital
period of two galaxies tends to be millions of years – much longer than can be measured
on human time scales! To determine masses of objects in the Solar System awaited to two
advances beyond Newton: determination of distances within the Solar System (i.e. beyond
the Moon, which the Greeks knew) and the calibration of “big gee” (G) the Gravitational
Constant.
1.
Distances within the Solar System
Kepler’s Third Law, P 2 = a3 , related something we could determine, sidereal periods
of planets, to something we wished to know, a, the semi-major axis (or size) of their orbits.
Unfortunately, the relation was in terms of the Earth’s value for a, which is called the
Astronomical Unit, or AU. The AU is defined as the semi-major axis of the Earth’s orbit.
But how can we determine it? (Note that because the Earth’s orbit is nearly a circle, the AU
is also nearly the mean distance of the Sun from the Earth. However, the precise definition
of the AU is, as given above, the value of the semi-major axis of the Earth’s orbit).
One approach to calibrating the AU in physical units such as meters, would be to
determine the distance to the Sun directly at any given time. However, this turns out not to
be easy to do. A simpler method is to get some other distance within the Solar System to a
planet (or asteroid) that obeys Kepler’s Third Law and effectively then, one has determined
the scale of the Solar System. That is, we know instantaneously the distance between any
two objects orbiting the Sun from Kepler’s Laws, and if we can get the actual distance (in
meters, not in AU!) between these objects then we can calibrate the AU.
This was first done by Giovanni Cassini in 1672. His method is known as the “parallax”
method of distance determination and is widely used in astronomy even today, although not
for Solar System objects. He used observations of the planet Mars obtained simultaneously
from two locations on the Earth that were greatly spaced in distance. One was in Europe and
the other in South America. Mars appeared to be in slightly different directions against the
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fixed stars when viewed from these two different locations on Earth. By measuring the angle
through which Mars appeared to move when viewed from the two different locations, knowing
the distance between those locations on Earth, Cassini could employ simple geometry to get
the distance to Mars. He achieved an accuracy of about 7% and calibrated distances within
the Solar System (i.e. calibrated the AU in terms of meters).
Today we have much more accuracy in our value of the AU. Distances to planets are
routinely determined by bouncing radar signals off of them and timing how long the signal
takes to return to Earth.
2.
Calibrating the Gravitational Constant
To make progress in determining masses of objects, we must first directly measure how
strong gravity is. That is we must determine the value of ”big gee” (G), the ”Universal”
constant of gravity in Newton’s equations. Surprisingly, this was not done until well after
Sir Isaac Newton’s work. It was first accomplished by the British physicist Lord Cavendish
(see link on course page). He used a very sensitive instrument called a torsion balance to
measure the force exerted by large masses on each other. His experiment yielded the mass
(and, therefore density) of the Earth for the first time. Once G was known, it was possible
to use ”little gee” (g), the gravitational acceleration at the surface of the Earth to determine
M, the mass of the Earth. From Newton’s theory, we have
GMm
F = ma = mg =
R2
where g is the acceleration of an object at the surface of the Earth and R is the radius of
the Earth and M is the mass of the Earth. From this, it follows that
GM
g= 2
R
so that with g known (9.8 m/s/s) and R known and G now known, M is determined. From
the mass of the Earth, its mean density (ρ) follows immediately as its mass divided by its
volume. Since the Earth is a sphere (to a good approximation!) its volume is 34 πR3 so
ρ=
3.
3M
= 5.5 gm/cm3 .
3
4πR
Two-Body with m << M
For objects besides the Earth we normally cannot measure their surface accelerations
directly, so we need to use their gravitational affect on nearby objects (e.g. moon, planet,
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companion star, etc.) to determine their mass. The simplest case to analyze and determine a
mass for is when m << M and the motion of the large object can be ignored. Note that since
the motion of m is independent of its actual mass, this method can only be used to determine
M, not m. In reality, the method determines the sum of the masses M+m, but since m <<
M this is essentially the same as determining M. The relevant equation is Kepler’s Third
Law as formulated by Newton, namely
P2 =
4π 2 3
a.
GM
It is clear from this equation that if we can measure P and a we can solve for M, because
the other quantities in the equation are known constants.
Normally, P is pretty easy to determine for any cyclical motion just by watching it. For
example, we can determine the period of the Moon around the Earth just by watching its
motion against the fixed stars. We can get P for Earth orbiting the Sun from the time it
takes the Sun to go around the sky once. The orbital period of other planets is derived from
their Synodic period (S) as described in an earlier chapter. As an example, the mass of the
Sun might be found from this equation by using the data for the Earth’s orbit. We know
that the period is 1 year (or 3.16 × 107 sec) and we know that the AU is 150 million km
(or 1.5 × 101 1 cm). Therefore, we know all quantities in the equation above except M, the
mass of the Sun, and we can solve for it, being careful to use values with the same units.
Using the AU in cm and P in sec, we need G in cgs units, and its value is G = 6.67 × 10−8 .
Plugging in, we find the mass of the Sun to be about 2 × 1033 gm.
Once a mass is determined for an object, its mean density again follows easily, assuming
we know how big the object is. The diameter of the Sun is easy to determine from Earth, in
part by the fact that the Moon happens to just fit right over it during a solar eclipse! Hence,
the Moon and Sun have close to the same angular size, which is about one-half of a degree
or 30 arc-minutes. Knowing the angular diameter of the Sun and its actual distance (the
AU) allows us to calculate its actual diameter in terms of meters and, therefore, its radius.
From its radius, and the fact that it is a sphere (as closely as we can tell!) its volume follows.
From its mass and volume, its mean density follows, and is about 1.4 gm/cm3 . Note that the
mean density of the Sun is much smaller than that of the Earth which, of course, reflects the
fact that its composition is much different. The Sun is primarily made of Hydrogen, while
the Earth is primarily made of Oxygen (bound to silicon, known as silicates or “rocks”).
Masses of planets with moons are very easy to determine by this method. A good
example is Jupiter, which has 4 easily visible moons, discovered by Galileo. By observing
any one of the moons’ motion one can determine its period and its maximum separation
from Jupiter, in terms of angle. The period can be converted to seconds and the angular
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separation can be converted to a size of orbit (around Jupiter) using the known distance
to Jupiter. Hence, with P and a known for that moon, we can use Kepler’s Third Law
(Newton’s form) to determine the value of M, which is here the mass of Jupiter, since it is
the central massive object about which its moons are orbiting. Again, once the mass of the
planet is found, we can calculate its mean density. For Jupiter this turns out to be close to
that of the Sun, about 1.33 gm/cm3 . This is an indication that Jupiter is a much different
kind of planet than the Earth – being composed primarily of Hydrogen, like the Sun, not
oxygen and heavier elements like the Earth.
This method can be used to determine the masses of any planets (or even some nonplanets) that have moons orbiting them. This applies to all the planets now except Mercury.
Venus also does not have a moon but it does have an artificial satellite (which we put there)
orbiting it. Some asteroids even have moons of their own and this method can be used to get
their masses. A famous non-planet known as Pluto has a moon and its mass (and density)
can therefore be determined. Densities of objects in the Solar System are (note that liquid
water and pure ice have a density of about 1 gm/cm3 and typical rocks on the Earth’s surface
have a density of about 3 gm/cm3 : Sun (1.4), Mercury (5.4), Venus (5.2), Earth (5.5), Moon
(3.4), Mars (3.9), Jupiter (1.3), Saturn (0.7), Pluto (2.0). Based on this we can divide object
in Solar System into those composed primarily of the light elements Hydrogen and Helium
(Sun, Jupiter, Saturn), those composed of “rocks” including silicates and iron (Earth, Venus,
Mercury), rocky planets with less iron (Moon, Mars) and icy objects (Pluto).
4.
General Two Body Orbits
Often it turns out that the difference in masses between the two orbiting objects, m
and M are not that different, in which case M must be replaced by m+M and a must be
replaced by the sum of the semi-major axes in Kepler’s Third Law. So, knowing the period
and total size of the orbit only gives the sum of the masses, not the individual masses. To get
individual masses we must get the relative sizes of the orbits of the two objects, remembering
that they will both orbit the center of mass. They obey the relation that m1 a1 = m2 a2 . It is
generally the case for binary stars that we need to know the motions of both stars in order
to determine the individual masses. A beautiful example of this effect, where the motion of
the more massive object must be taken into account, is the discover of “exoplanets”, planets
around other stars (see link on course page).
Planets do exert a gravitational influence on their host stars and so their presence can
be dettected by the small orbit, or “wobble” that the central star undergoes as it responds
to the gravitational pull of the planet. This motion may be very small (only meters/second
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in terms of velocity), yet it can be detected with modern techniques. Hundreds of planets
have now been detected around nearby stars by this method. From the amplitude of the
motions of the stars we can determine the mass of the object they are orbiting – i.e. the
planets. So far, this method has detected objects as small in mass as Neptune (about 15
Earth-masses), but it is not capable of finding Earth-sized planets.
5.
Unbound Orbits
It is also possible to determine the mass of an object with nothing orbiting it, just by
the gravitational deflection it causes on an object passing close to it. In other words, one can
use the shape of the unbound (hyperbolic) orbit to get the mass of the deflecting objecting.
This is how the mass of Mercury must be determined, since it has not orbiting satellites,
even artificial ones, however we have done “fly bys” of satellites passing close to it along
unbound orbits. This is also how we can determine the masses of moons of other planets
(e.g. Jupiter) by having a satellite pass close to the Moon and seeing how much deflection
there is). We can also get masses of asteroids in this way as our probes fly by them by
measuring the amount of deflection of the orbit.
6.
Perturbations
The gravitational interactions between the smaller objects (e.g. planets) orbiting a
larger object (e.g. star) produce so-called perturbations to the elliptical orbits that the
planets would otherwise follow. This effect can be quite pronounced and add up over time to
change the characteristics (e.g. eccentricity) of orbits. It was actually key to the discovery
of a new planet in our own solar system – Neptune. The planet Uranus was discovered
entirely by chance in the late 1700’s by Sir William Herschel. When its motion was followed
carefully, it was found not to be obeying Kepler’s Laws precisely. This was attributed to
the existence of an additional planet, as yet unseen. In the mid 1800’s the exact location
of Neptune was calculated using the perturbation effect it must be having on Uranus. The
planet was then discovered in one night at very close to its predicted position. This was
a triumph for Newton’s law of gravity and for science in general. In some cases, the mass
of an orbiting object can be determined by the amount of perturbation it produces in the
orbit of another object. Pluto was originally thought to be having a significant perturbing
effect on Neptune, making it much more massive than we now know it is. This was leading
to the absurd conclusion that the density of Pluto was so high that it must be made of
gold! (or something like that). We now know that the small discrepancies thought to be
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measured in Neptune’s orbit and attributed to Pluto’s influence were actually only noise in
the measurements.
7.
Non-Keplerian Motions, Virial Theorem and Dark Matter
We will come back to this later in the course, but it is worth mentioning here that this
same concepts of orbits of objects around each other or motion in response to the presence of
another mass is the central idea throughout astronomy in determining mass. For example,
stars orbiting around the center of a galaxy can be used to determine the mass of the matter
inside their orbiting position. The discovery of rapidly orbiting stars near the center of our
own galaxy led to the realization that there is a central black hole there. The fact that
orbital speed increases (or stays the same) as one moves further out in a galaxy, rather than
decreasing, as in the solar system, led to the discovery of “Dark Matter” in galaxies (not
the same as black holes). The Virial theorem states that twice the total kinetic energy in a
gravitationally bound cluster (of stars or galaxies) is equal to the total potential energy. This
concept is used to measure masses of whole clusters of stars and galaxies and also indicates
that there is much more mass than expected based on the visible light. This is additional
strong evidence for dark matter in the Universe. We will come back to these concepts later
in the course but wanted to mention them here because they are all related to the same basic
idea of how mass is determined, that starts with Newton’s laws.