Expert: Philip Stahl - 11/28/2006

Question
Is the Sun, our sun, moving through space and, if so, what observable, demonstrable evidence do we have for it?

Answer
Hello,

Offhand I will bet when you asked this question you thought it might have a simple and straightforward answer. Alas, this is not the case. Much of this is technical info, but I have omitted explaining details as this would have expanded the answer to the size of a small book. So, I am hoping you will go ahead and 'google' any unfamiliar terms!

Anyway, yes, our Sun is moving through space – in particular the motion of the solar “apex” at about 19.5 km/s. To deduce this we use the averaged proper motions obtained from the nearby field of stars, as follows:

Q  x

b


P   x  ------------------------------------------ d--------S



Here, we regard the Sun’s motion from P to Q and a star at S (considered “fixed” ) for the moment.

A star regarded to lie in the heliocentric position PS at the beginning of the year, lies in direction QS at the end. It may therefore be said to have exhibited a proper motion u, given by:

/_ PSQ

and we may write:

sin u/ b  = sin (180 – phi 2)/ d

or sin u = b/ d (sin phi 2)

(Note: If you imagine a transversal drawn from S to Q, the angle phi 1 is /_  SPQ and phi 2 is the alternate interior angle. The parameter b is the distance between P and Q)

As it turns out, u is such a small angle in practical terms that the suffixes 1,2 can be omitted since either phi 1 or phi 2 may be used in the equation, whence:

sin u =  (b/ d) sin phi  

(Say, phi = phi 1 =    /_SPQ)

we have also:

b = n V

where n is the number of seconds in a year, and V is the space velocity

The star’s parallax P =  a/ d

where a is the semi-major axis of the Earth’s orbit.

combining the previous equations, we obtain:

sin u / sin P  =   (nV/ a)  sin phi

or

u = VP sin phi /  (4.74)

(after inserting values for n and a)

The preceding equation shows that if the Sun had a space velocity V, not only would the stars exhibit proper motions even if at rest, but that the star’s proper motion would be proportional to its parallax and also proportional to the sine of the angle its direction makes with the direction the Sun is moving.

Now, all the above are measurable, including the parallax P, n and a.



Now, in assessing the actual solar motion – the technical sub-branch of astronomy known as astrometry is used. This branch is concerned with the very exact computation of star positions, motions, and preparation of star catalogues.

The central problem for astrometry is that – given an observed proper motion for some star in its right ascension:

u (ar) =  u’(ar) +  u(a)

we can isolate the proper motion arising from the star’s own motion (first term, right side) from the proper motion arising from the solar motion.

In other words, to the extent that we can obtain u(a) we can establish that the Sun itself is doing the moving in a specific direction.

The technical problem may be boiled down to subtracting out from N total observed proper motions, the systematic parallel proper motions imposed on the N stars by the solar motion.

In practice, we may have N = 1000 for which we get:

u(a1) + u(a2) + u(a3) + ………u(aN) = (u’ (a1) + u’(a2) + ……u’(aN) + N x u(a)

Now, because of the random nature of the stars’ own velocities (as many going in direction X say as in (-X) the quantity in brackets will ne small in comparison with
N x u(a). Hence,

u(a)  =  1/ N [u (a1) + u(a2) + ………u(aN)]

Similarly, if u(d) is the component of parallactic proper motion in declination for the collective of stars:

u(d) = 1/ N [u(d1) +  u(d2) + …………u(dN)]

Thus, for any region of stars whose center has known coordinates a and d (right ascension and declination) values may be found for u(a) and u(d) the components of parallactic proper motion. By observing different regions of the sky and using the same procedure, a final value for the solar apex can be found with minimal errors.

The magnitude of the solar speed itself is also found from the set up.  The component of the Sun’s own velocity V along the heliocentric direction (PQ from earlier) is

V cos (phi)

This must produce on each star in the selected measurement region a parallactic radial velocity of

(- V cos (phi))

Let Vr be the observed radial velocity of the rth star and  Vr” the radial velocity caused by its own space velocity. Then:

Vr =   Vr” – V cos(phi)

summing over all N stars of the region we obtain:

V1 + V2  + Vr +  …..+   VN  =   (V1’  +  V2’   +  …+  Vr’  +…..+ VN’) – NV cos (phi)


yielding:

V =  -(1 / N cos (phi)) [V1 + V2 + ……..Vr + ……..VN]

Again the V1, V2 quantities in the bracket largely cancel out owing to the random distributions of magnitudes.

The value then found for V is 19.5 km/s

It is also possible, by the way, to find the *position* of the solar apex from radial velocity statistics – but that adds another layer of complexity I prefer not to get into here!

Be careful what you ask for!  :}