In history we are taught that the Catholic Church was wrong, because the Sun does not move around the Earth, instead the Earth moves around the Sun.
But then in physics we learn that movement is relative, and it depends on the reference point that we choose.
Wouldn't the Sun (and the whole universe) move around the Earth if I place my reference point on Earth?
Was movement considered absolute in physics back then?
Imagine two donut-shaped spaceships meeting in deep space. Further, suppose that when a passenger in ship A looks out the window, they see ship B rotating clockwise. That means that when a passenger in B looks out the window, they see ship A rotating clockwise as well (hold up your two hands and try it!).
From pure kinematics, we can't say "ship A is really rotating, and ship B is really stationary", nor the opposite. The two descriptions, one with A rotating and the other with B, are equivalent. (We could also say they are both rotating a partial amount.) All we know, from a pure kinematics point of view, is that the ships have some relative rotation.
However, physics does not agree that the rotation of the ships is purely relative. Passengers on the ships will feel artificial gravity. Perhaps ship A feels lots of artificial gravity and ship B feels none. Then we can say with definity that ship A is the one that's really rotating.
So motion in physics is not all relative. There is a set of reference frames, called inertial frames, that the universe somehow picks out as being special. Ships that have no angular velocity in these inertial frames feel no artificial gravity. These frames are all related to each other via the Poincare group.
In general relativity, the picture is a bit more complicated (and I will let other answerers discuss GR, since I don't know much), but the basic idea is that we have a symmetry in physical laws that lets us boost to reference frames moving at constant speed, but not to reference frames that are accelerating. This principle underlies the existence of inertia, because if accelerated frames had the same physics as normal frames, no force would be needed to accelerate things.
For the Earth going around the sun and vice versa, yes, it is possible to describe the kinematics of the situation by saying that the Earth is stationary. However, when you do this, you're no longer working in an inertial frame. Newton's laws do not hold in a frame with the Earth stationary.
This was dramatically demonstrated for Earth's rotation about its own axis by Foucalt's pendulum, which showed inexplicable acceleration of the pendulum unless we take into account the fictitious forces induced by Earth's rotation.
Similarly, if we believed the Earth was stationary and the sun orbited it, we'd be at a loss to explain the Sun's motion, because it is extremely massive, but has no force on it large enough to make it orbit the Earth. At the same time, the Sun ought to be exerting a huge force on Earth, but Earth, being stationary, doesn't move - another violation of Newton's laws.
So, the reason we say that the Earth goes around the sun is that when we do that, we can calculate its orbit using only Newton's laws.
In fact, in an inertial frame, the sun moves slightly due to Earth's pull on it (and much more due to Jupiter's), so we really don't say the sun is stationary. We say that it moves much less than Earth.
(This answer largely rehashes Lubos' above, but I was most of the way done when he posted, and our answers are different enough to complement each other, I think.)
yes, you may describe the motion from any reference frame, including the geocentric one, assuming that you add the appropriate "fictitious" forces (centrifugal, Coriolis, and so on).
But the special property of the reference frame associated with the Sun - more precisely, with the barycenter (center of mass) of the Solar System, which is just a solar radius away from the Sun's center - is that this system is inertial. It means that there are no centrifugal or other inertial forces. The equations of physics have a particularly simple form in the frame associated with the Sun. $$ M_1 d^2 / dt^2 \vec x = G M_1 M_2 (\vec r_1-\vec r_2) / r^3 + \dots $$ There are just simple inverse-squared-distance gravitational forces entering the equations for the acceleration. For other frames, e.g. the geocentric one, there are many other inertial/centrifugal "artificial" terms on the right hand side that can be eliminated by going to the more natural solar frame. In this sense, the heliocentric frame is more true.
This was going to be a comment on Luboš Motl's answer, but it would be more appropriate as a full answer now.
His answer says: Laws of physics can be written more simply for the solar system's center of mass (barycenter) than for a point on Earth (geocentric).
Just one thing! One mustn't neglect the non-idealities of the barycenter itself, which has a location in the Milky Way that biases it gravitationally at least. On the surface this is splitting hairs, but the greater point is that the idealness of any reference frame is also relative, and no "ultimate" frame exists.
Likewise, choosing a point on the skin of an elephant over a geocentric point is sacrificing universality just as much as choosing a geocentric point over the barycenter is. To a flee however, consideration of physics formulated at a point beyond the surface of the elephant may be just "academic". Sound familiar?
Yes, the proposition: "the sun moves around the earth" had the earth immobile. This suited the theology of the times which was completely anthropocentric and that is why it prevailed over other theories coming from antiquity, like Aristarchos', who had a heliocentric proposal.
The relativity of motion was explored, as Lubos describes, when equations could be written down, and one chooses the heliocentric for its beauty and simplicity. The epicycles exist if one plots the solutions in a geocentric system, but they are so cumbersome and "ugly" as a shorthand of physics.
There may be a confusion : it is wrong to say that the Earth is the centre of the Universe, that is, the (unique) point from which the Universe is to be (fundamentally) described (the fact that the Sun turns around the Earth is only a consequence of this) ; what actually matters is that there is no centre of the Universe : there is no such point ; the description of the Universe from any point is equivalent to the description of the Universe from any other (then you are allowed to describe motions either from the Earth or from the Sun).
Mathematically, in classical mechanics, the Universe is said to be an affine space.
Both Sun and Earth move in circles around their barycenter i.e. centre of mass.
The trick is that since Sun is too massive, the center of mass is too close to the sun, actually beneath the surface of the Sun, which makes the motion of Sun negligible. And, we say that Earth moves around the Sun.
I have to use this as a chance to repeat a great story about the philosopher Wittgenstein, related by his student Elizabeth Anscombe:
[Wittgenstein] once greeted me with the question: "Why do people say that it was natural to think that the sun went round the earth rather than that the earth turned on its axis?" I replied: "I suppose, because it looked as if the sun went round the earth." "Well," he asked, "what would it have looked like if it had looked as if the earth turned on its axis?"
But what about physics? In terms of actual physical theories, does the sun really go around the earth, or does it only appear to do so because we're viewing it from the rotating reference frame of the earth?
A rotating frame is distinguishable from a nonrotating frame, without reference to anything external. This is true both in Newtonian mechanics and in special and general relativity. There are various ways to tell if you're in a rotating frame, including a Foucault pendulum, a mechanical gyroscope, or a ring-laser gyro of the type used in commercial jets. The Foucault pendulum as a proof of the earth's rotation dates back to about 1850. (Long before then, heliocentrism had become accepted among physicists on less definitive grounds, such as the fact that Kepler's laws have a simple form in a heliocentric frame.) As a relativistic example, the analysis of the famous Hafele-Keating test of general relativity required the introduction of three effects: kinematic time dilation; gravitational time dilation; and the Sagnac effect, which is sensitive to the rotation of the earth.
There are other theories in which you can't detect a frame's rotation except relative to distant matter, e.g., Brans-Dicke gravity. The original paper on B-D gravity is available online http://loyno.edu/~brans/ST-history/ and is very readable even if you're not a specialist. The positive results from the techniques listed above would then be interpreted not as evidence of absolute rotation but as evidence of rotation relative to distant galaxies. But B-D gravity is no longer viable based on solar-system tests dating back to the 1970's. So if you like, you can say that Galileo was only finally proved right in the 1970's.
The sun, moon, earth (and so on) all move around each other.
The reason we say the earth moves around the sun is because the effects are more visible on a macro scale, and easier to predict with reasonable precision. Yes, it's most correct to say that all motion is relative, but it gets a lot more complicated to explain it if you're speaking to a layman.
Since it is a recurrent question, I have rather to add my answer here than to more recent ones.
I hope I can make more clear some points which was not completly well focused in some previous answers.
Kinematic description
Once we have chosen whatever reference frame we like (here does not matter if inertial or not) and we have a description of the trajectories of N bodies, say N vectors ${\bf r}_i(t)$, we can always use a reference frame centered on one of the bodies, say the a-th, by simply subtracting the position vector of the chosen body to any other position vector. Therefore, in this new reference frame the trajectories of the original system of N bodies will be: $$ {\bf r}^{\prime}_i(t) = {\bf r}_i(t) - {\bf r}_a(t).~~~~~~~~~~~~~~~[1] $$ It is clear that in this new frame ${\bf r}^{\prime}_a(t)=0$ by construction, i.e. the a-th body is at rest forever.
An example of such transformation of coordinates is the change of reference frame required if we want to find the proper description of the Solar system as seen by an observer on the Earth, starting from trajectories in the (inertial) reference frame where the center of mass of the Solar system is at rest. Notice that an observer at rest on the surface of Earth is not only translating with the planet, with respect to the center of mass, but she/he is also rotating, so the actual transformation would be more complicate than eqn. [$1$]. However we can ignore the need of an additional rotation of our vectors if we confine our considerations to reference frames which do not rotate relatively to the original frame.
At this point it should be clear that there is nothing wrong to describe the motion of the Solar system bodies from the Earth. It is just one of the infinite possible choices of the origin of the reference frame, an probably the most useful for Earth based observers. It has the same right to be used as a reference frame fixed on a moving car to describe what passengers see.
However, the possibility of changing point of view, does not imply that different choices would provide the same description of the trajectories in an N-body system. Quite interestingly, if we start from a reference frame where body $a$ is at rest, i.e. ${\bf r}_a(t)=0$, where a second body b moves according to ${\bf r}_b(t)$, and we move to a new reference frame based on body $b$, in the new system body $a$ will be described by the vector ${\bf r}^{\prime}_a(t)=-{\bf r}_b(t)$. This implies that motion of $a$ as seen by $b$ or motion of $b$ as seen by $a$ differ only by an inversion and therefore they have the same synthetic description.
What about applying the above consideration to the Earth-Sun system? In the case of the two-body system things are quite simple. The shape of Earth trajectory as seen from Sun or that of Sun as seen from Earth are the same. In addition, since the center of mass of the Sun-Earth system is within Sun, Earth trajectory as seen from Sun is almost coinciding with the same orbit as described from the center of mass.
In the next two figure I have plotted the orbit of the two bodies in the reference frame of the center of mass
and in the reference frame of (non-rotating) Earth. Distance units are millions of kilometers.
Things change a lot when we describe motion of other Solar system bodies. THe next two plots show the motion of Sun, Venus, Earth, Mars and Jupiter, as seen from the center of mass of the system or from (non-rotating) Earth.
Even at this kinematic level, the greater simplicity of the description in the center of mass frame is evident. Nevertheless, I want to stress once more that nothing is wrong with this description. It is the closest to what we get frome Earth-based observations.
Dynamic description
From the point of view of solving a problem of Newtonian dynamics, we all know that the center of mass reference frame of an N-body system is convenient. Since it is an inertial frame, we can use Newton's law ${\bf F}=m{\bf a}$ in connection with Newton's gravitional force law, without need of introducing additional inertial forces.
Notice however that once one has written the set of differential equations of motion for the gravitational N-body problem: $$ {\bf a}_i = G\sum_{j\neq i} m_j \frac{({\bf r}_j - {\bf r}_i)}{\left|{\bf r}_j - {\bf r}_i\right|^3} $$ it is trivial to write down the equations of motion referred to body $a$: $$ {\bf a'}_{i} = G\sum_{j\neq i} m_j \frac{({\bf r'}_j - {\bf r'}_i)}{\left|{\bf r'}_j - {\bf r'}_i\right|^3} - G\sum_{j\neq a} m_j \frac{({\bf r'}_j - {\bf r'}_a)}{\left|{\bf r'}_j - {\bf r'}_a\right|^3} ~~~~~~~~~~~[2] $$ where $$ {\bf r'}_i = {\bf r}_i - {\bf r}_a $$ and $$ {\bf a'}_i = {\bf a}_i - {\bf a}_a = \frac{{\mathrm d}^2 ({\bf r}_i - {\bf r}_a ) }{{\mathrm d}t^2}. $$ There are two interesting things to notice in eqn. 2, the first one could help to clarify some statements present in other answers:
A very late answer, one that I hope adds to the excellent answers by Mark and Luboš.
From the perspective of Newtonian mechanics, there's nothing wrong per se with using a geocentric point of view. Such a point of view does require adding fictitious forces and torques that would otherwise be absent in an inertial perspective, but if makes sense to do that, that's okay. That said, there's a world of difference between choosing to use a geocentric perspective when doing so makes sense such as predicting the weather compared to a now non-scientific mandate that one must always use a geocentric perspective. There's a nice explanation of those fictitious forces and torques that result from choosing to use a geocentric perspective: They're a fiction that results from that choice of perspective. This mandate would instead somehow make all of those fictitious forces and torques real. What makes these forces occur, and why in the world do they disappear when we choose to look at things from a different perspective?
Even though a geocentric perspective is conceptually valid from a Newtonian perspective, the concept of parsimony (aka simplicity, aka Occam's Razor) says we must reject the idea of returning to a mandated geocentric point of view (and thereby forgo half a millennium of scientific progress). Parsimony has played a very important role in science since Galileo's time. Scientists much prefer simple explanations over complex ones. Using a geocentric perspective to describe the motion of an exomoon about an exoplanet is a ludicrous proposition.
From the perspective of general relativity, there is something wrong per se with using a geocentric point of view to describe the entire universe. While coordinate systems are global in Newtonian mechanics, they are local in general relativity. Coordinate systems are local charts on Riemannian space-time in general relativity. They do not have universal extent. A mandated geocentric perspective does not make sense in terms of general relativity.
It could be deduced with some careful observations, a bit of logic, and the premise that the simplest solution is most likely the correct one.
We can make observations of stars and planets, and use parallax to estimate their distances, and reasonably conclude that the stars are very far away, and that the Sun is very big and probably very heavy. We can also deduce (the way the ancient greeks did) the size of the Earth and note that the Sun is much, much bigger.
We can take a ball on a string and spin it around, and observe that we need some amount of force to keep the ball constrained to a circular path.
If we were to assume that the Earth was stationary and everything in the sky was spinning around us, what would be keeping it all in place (think about the ball on the string)? On the other hand, if the Earth was spinning instead, there is no requirement for anything to keep everything up there from flying off in all directions. So, the Earth is spinning, not the heavens.
The paths of all the planets make much more sense if they are seen to be travelling around the Sun rather than the Earth - with that retrograde motion of Mars and all. And it would look very much like what we see when we look very carefully at Jupiter and its moons - big body in space will smaller ones orbiting around it.
So, you could conclude that all the planets in the night sky orbit the Sun, but the Sun orbits the Earth. Except that the Sun is so much bigger, and the model would be so much simpler if the Earth was orbiting the Sun just like the other planets.
...And we've arrived at our current undertstanding.
In my opinion "description of trajectory" is not a topic of physics it is a topic of kinematics (geometry of motion, see http://en.wikipedia.org/wiki/Kinematics). Whereas explaining the mechanism which causes an object to follow a particular trajectory IS a matter for physics.
To say that "B goes around C" is to describe a trajectory in space and time. A trajectory may be described by graphical means e.g. a circle, an ellipse, a helix. But all such graphical presentations of a trajectory are subjective i.e. they depend on the frame of the observer. An observer attached to C will observe that B goes around C. Whereas an observer attached to B will observe that C goes around B.
In a Euclidean system of description a particular object trajectory can be described absolutely by relating the spatial displacements (distance and direction) of the object (at various moments in time) relative to one or more reference objects (whose trajectories are themselves known ... relative to some useful standard).
If you sit in an office chair and someone spins it around you will see the walls of the office move around you. I contend that it can be acceptable and useful to say that "the office moves around you". Likewise it is misleading to say categorically that "the office DOES NOT move around you". Any description of movement (motion) relates the positions of at least two objects. This applies to both linear and non-linear patterns of motion. Physicists may choose to describe, measure and account for the sensations and motions which you experience by choosing particular frames of reference because they are more simple or more useful. But this does not dictate how you choose to describe the dynamic geometry of your experience.
Therefore the following descriptions of dynamic geometry are all acceptable and potentially useful and potentially ambiguous:- "the Earth moves around the Sun" "the Sun moves around the Earth". "the Sun and the Earth each move around their mutual barycentre".
There are experimental evidences of absolute motion of the Earth around the Sun. There is a dipole anisotropy in fine measures of the Background Radiation temperature that is known from the analysis of the COBE satellite measures, in the early 90s. See for instance this paper.
In order to make the adequate corrections, so that the Cosmic Background Radiation "seems" isotropic, the absolute velocity of the Local Group against the Cosmic Background Radiation must be accounted for, but that correction depends on the month of the year, because a small part of the correction comes from the orbital speed of the Earth around the barycentre of the Solar System (among other terms).
That small part of the corrections needed is exactly what you would expect if you assumed that is the Earth who is going around the Sun, and not vice versa.
(the cosmic background dipole anisotropy, image from map.gsfc.nasa.gov)
Here is an extract from the abstract of the quoted paper:
We present a determination of the cosmic microwave background dipole amplitude and direction from the COBE Differential Microwave Radiometers (DMR) first year of data (...) The implied velocity of the Local Group with respect to the CMB rest frame is $v_{LG}=627 \pm 22 km s^{-1}$ toward (...). DMR has also mapped the dipole anisotropy resulting from the Earth's orbital motion about the Solar System barycenter, yielding a measurement of the monopole CMB temperature (...) $T_0=2.75 \pm 0.05 K$
This doesn't mean however, that there is an absolute reference frame in the Universe. Other comoving observers will detect another dipole anisotropy. The Last Scattering Surface, as well as the cosmological horizons are different for different comoving observers. But nevertheless it proves that it is the Earth that moves around the Sun, and not vice versa. Since the 90s this is no more a philosophical issue: WE are moving certainly, absolutely, surely and gloriously, around the Sun.
In ancient times, the mechanics of orbital motion due to gravitational attraction wasn't known. What was known, though, was that if Earth orbits Sun, then stars would display a cyclical motion called "parallax." The Greeks actually predicted this, but didn't have the technology to observe it. This was a main reason the geocentric solar system model stood for so long. The parallax is real though, and observable, and provides direct observational evidence that Earth orbits Sun.
I'm putting down a much shorter answer: The sun doesn't move, the earth (together with the other revolving planets in our galaxy) does. The earth basically rotates around the sun in a ring (and its axis, but that's besides the point). Besides, the churches have always made many wrong claims (especially in the middle ages), such as that the earth was flat. They though so because we 'stand on top of the earth' whilst there really is no up or down.
