The changing pole star

Many people know that Polaris, the brightest star in the constellation Ursa Minor (The Little Bear), is also known as the pole star. Indeed, the name Polaris itself was invented in the sixteenth century and is derived from the Latin stella polaris -pole star.

The location of Polaris – Image credit Wikimedia Commons

Polaris is located roughly one and a half times the diameter of the Moon away from the projection of the North Pole into the sky. This point is known as the North Celestial Pole and, to an observer in the Northern Hemisphere, Polaris appears almost exactly due north.  Over the course of a night, the Earth’s rotation means that all stars appear to rotate around the North Celestial Pole.  This includes Polaris as well, because it isn’t located exactly at the North Celestial Pole.

A long exposure image of the night sky looking northward. All the stars appear to move in arcs around a point close to Polaris, which itself moves in a small arc. The orange and green areas at the bottom of the image are caused by light pollution.

Because Polaris is almost exactly due north anywhere in the Northern Hemisphere, for many centuries, it has been used by navigators to find their way. Two bright stars in Ursa Major (The Great Bear) are known as “The Pointers” and can be used to locate Polaris.

Using ‘The Pointers’ to locate Polaris

The changing pole star

Although Polaris is almost directly above the North Pole today, it has been known for over two thousand years that the orientation of the Earth’s axis is not fixed with respect to the background stars. Instead, it slowly rotates in a circle, completing one revolution every 25 800 years.  This causes the position of the North Celestial Pole to gradually change.

Line AB shows the current alignment of the Earth’s axis, with respect to the background stars. Line CD shows the alignment of the Earth’s axis in 13 000 years’ time.

The current orientation of the Earth’s rotation axis is shown as the line running from the North Celestial Pole (marked A) to the South Celestial Pole (marked B). Over 25 800 years the projection of the Earth’s rotation axis traces out a circle of radius 23.5 degrees (roughly 45 times the apparent diameter of the Moon). This circle is centred at a point called the North Ecliptic Pole which lies at a right angle to the plane of the Earth’s orbit. 13 000 years from now in thee year 15000 the North Celestial Pole will be at the opposite side of this circle from its current location and there will be no bright north pole star.

The diagram above shows how the North Celestial Pole changes over the 25 800-year cycle. Five thousand years ago, it was very close to Thuban in the constellation Draco -the Dragon.

In the year 10 000 the North Celestial Pole will lie close to Deneb in the constellation Cygnus – the Swan. In the year 11 000 it will lie close to Delta Cygni in the same constellation. In 14 000, Vega which is the fifth brightest star in the sky will be close to the North Celestial Pole.

The change in the alignment of the Earth’s axis is due to an effect called precession. A more familiar example of precession is when a rapidly spinning top is tilted at an angle to the horizontal, the axis of the top will trace out a circular path as shown in the diagram below.

In the case of the top, precession is caused by gravity exerting a torque, which is a twisting force around the origin (marked as O in the diagram above). If the top is not spinning this torque will simply cause it to fall over. However, if it is rapidly spinning the torque acts to constantly change the orientation of the spin axis causing it to trace out a circular path.  (Technically speaking the torque, which is normally given the symbol τ is defined as the vector product of the gravitational force on the top with a line joining its centre of mass with the origin O. For more details see the notes at the bottom of this post.)

In the case of the Earth, the torque is exerted mainly by the Sun and the Moon and arises because the Earth is not a perfect sphere. It is slightly flattened having an equatorial bulge. The diagram shows that the pull of the Sun’s gravity on this bulge tries to change the orientation of the Earth’s spin axis in the same way that the Earth’s gravity tries to change the orientation of the top.

The Earth-Sun system viewed edge on

 

The south pole star

So far, this post has focused on the Northern Hemisphere pole star. There is currently no bright star near the South Celestial Pole. The nearest star visible to the naked eye is the star Sigma Octantis in the constellation Octantis (named after the octant – the navigational instrument). It lies about 1 degree away from the South Celestial Pole – roughly twice the apparent diameter of the Moon in the sky and is normally considered to be the southern pole star.  However, Sigma Octantis is so faint that it is only visible to the naked eye in rural areas away from light pollution.

Location of Sigma Octantis in the southern sky

 

However, the precession of the Earth’s axis means that, in the year 14 000, the second brightest star in the sky Canopus will lie within 8 degrees of the South Celestial Pole.

Movement of the South Celestial Pole around the South Ecliptic Pole over the 25 800-year cycle.

 

Polaris the multiple star

Polaris has a magnitude of 1.98 (which makes it the forty eighth brightest star in the sky) and lies at a distance of around 430 light years. Although it appears to the naked eye as a single star,  the Polaris star system consists of at least three and more likely four or five separate stars. The main star called Polaris A is actually two separate stars so close to each other that they cannot be resolved with Earth-based optical telescopes. The brightest component of this pair Polaris Aa is a  bright variable supergiant star which is on average 1360 times more luminous than the Sun. Its  close companion is called Polaris Ab and these revolve around each other once every 29 years. They have a more distant companion called Polaris B which takes 5000 years to do a single orbit around Polaris A.

In 1894 the American astronomer Burham discovered two faint stars  in the neighbourhood of Polaris which are called Polaris C and Polaris D. It is still unclear whether or not these are part of the Polaris system. It is possible that, although they lie close to Polaris in the sky, they may be foreground or background objects. However, this is unlikely. The chances of finding two stars of their brightness lying so close to Polaris and not being associated with  Polaris is less than 10%  for each star (Wielen et al 2000).  This means that the probability of at least one of them being associated with Polaris is around 99%. If they are associated with Polaris, then they are so far away that they will take around 100 000 years to complete one orbit.

Image credit NASA

And finally…

I hope you have enjoyed this post. If you are interested in finding out more about why the Earth’s axis precesses I have put some additional notes below.

 

 

 

Additional Notes

If we look again at the simpler case of the spinning top, discussed in the main article, the torque caused by the force of gravity causes it to precess. The rate of precession about the line OZ is given by the formula:

Where

ωp  is the angular velocity of the precession measured in radians per second. To convert from revolutions per second to radians per second multiply by 2π.

  • m is the mass of the top
  • g is the acceleration due to the Earth’s gravity
  • d is the distance between the centre of mass of the top and the origin
  • I is a quantity known as the moment of inertia.

 

  • ωp is the angular velocity of the precession measured in radians per second.

The derivation of this formula is normally covered in the first year of an undergraduate physics course and I will not repeat it here.  However, for any readers wishing to find out more, the following links are useful.

https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/11-3-precession-of-a-gyroscope

http://hyperphysics.phy-astr.gsu.edu/hbase/top.html

There is also an interesting video which describes in simple terms how precession works without using any mathematics.

https://youtu.be/EJWIl4MYMbw

 

Example

If we consider a top having: diameter of 5 cm, mass 80g, rotating at  a speed of 100 times a second and the distance between the bottom of its spindle and the centre of mass of 15 cm.

So lugging these values into the formula above

  • m = 0.08 Kg,
  • g = 9.8 metres /sec2
  • d= 0.15 metres
  • ω = 2π x 100 ≈ 628 radians /sec
  • I= ½ x 0.08 x 0.0252 =   2.5 x 10 -5  kg metres 2

gives a precessional angular velocity ωp of 7.5 radians /sec or, dividing by 2π, 1.2 revolutions per second.

In the ideal case, if there were zero friction at the bottom of the spindle and zero air resistance, then the top would continue to spin and precess at the same rate indefinitely. In reality, this is not the case, the top will gradually slow down as it loses rotational energy, start to become unstable and then eventually fall over.

 

 

 

Precession of the Earth’s axis.

As stated before, in the case of the Earth the torque is exerted mainly by the Sun and the Moon and arises because the Earth is not a perfect sphere and is slightly flattened having an equatorial bulge. The diagram shows that the pull of the Sun’s gravity on the bulge tries to change the orientation of the Earth’s spin axis so that it is at a right angle to the plane of its orbit. Because the Earth is spinning this torque will cause its axis to precess for the same reason that the spinning top precesses.

However, unlike the simpler case of the spinning top when the torque due to the Earth’s gravity remains constant, the torque due to the Sun on the Earth  varies throughout the year

  • It is strong in June (marked A in the diagram) when the North Pole is pointing towards the Sun and the centre of the bulges on either side of the Earth deviate greatest from the line joining the centre of mass of the Earth to the centre of mass of the Sun. This is shown as a dotted line in the diagram.
  • It is also strong in December (marked B) when the South Pole is pointing towards the Sun and the centre of the bulges on either side of the Earth deviate greatest from the line joining the centre of mass of the Earth to the centre of mass of the Sun. This is also shown as a dotted line in the diagram.
  • At the equinoxes in September and March (marked C and D) the centre of the bulges on either side of the Earth lie on the line joining the centre of mass of the Earth to the centre of mass of the Sun. In this case the net torque is zero.

It can be shown mathematically that the strength of the torque of the Sun on the Earth varies as the inverse cube of the distance between the Earth and the Sun. Because the Earth moves in an elliptical orbit and is closest to the Sun in early January and furthest away  in early July, this also causes an additional  variation in the torque resulting in the torque at the December solstice being stronger than it is at the June solstice.

The Moon’s orbit around the Earth is inclined at an angle which varies between 18.3 and 28.6 degrees to the Earth’s rotation axis, marked as θ in the diagram below. This means that the Moon also exerts a torque on the Earth. Even though the Moon’s gravitational pull is much weaker than that of the Sun its proximity to the Earth means that the average torque is approximately twice that due to the Sun.

  • It is strong when the Earth’s South Pole is pointed towards the Moon (marked A in the diagram) and the centre of the bulges on either side of the Earth deviate greatest from the  line joining the centre of mass of the Earth to the centre of mass of the Moon. This is shown as a dotted line in the diagram.
  • It is also strong 13.7 days later when North Pole is pointed to towards the Sun and the centre of the bulges on either side of the Earth deviate greatest from the line joining the centre of mass of the Earth to the centre of mass of the Moon. This is also shown as a dotted line in the diagram.
  • At the intermediate points (marked C and D) the centre of the bulges on either side of the Earth deviate lie on the line joining the centre of mass of the Earth to the centre of mass of the Moon. In this case the net torque is zero.

Because the Moon moves in an elliptical orbit, this also causes a variation in torque. In fact, as the  eccentricity of the Moon’s orbit around the Earth (which averages around 0.052) is greater than that of the Earth’s orbit around the Sun (0.0167), this variation is much greater

(The eccentricity, which is usually given the symbol e is a measure of how elliptical an ellipse is. It is defined as e2 = 1 – (b2/a2) where a is the long axis and b is the short axis of the ellipse).

A further complication is that the eccentricity of the Moon’s orbit isn’t fixed but varies between 0.0255 to 0.0775

Variation of the Moon’s orbital eccentricity – adapted from (Espenak 2012)

 

Although the torques due to the Sun and the Moon are the main factors in the Earth’s axial precession there are other elements which need to be taken into account. In particular, the torques from the planets, especially Venus which can approach as close as 38 million km to Earth. Because of the complexity and number of other factors contributing to the total torque on Earth it is not possible to calculate the rate of precession exactly and, in any case, it fluctuates over time. The current value from astronomical observations is that the Earth’s axis completes a full circle every 25 771 years.

 

References

Espenak, F (2012) Eclipses and the Moon’s orbit, Available at: https://eclipse.gsfc.nasa.gov/SEhelp/moonorbit.html (Accessed: 20 September 2020).

Wielen B, Jareiss H, Dettbarn C, Lenhart H, Schwan H (2000) Polaris: astrometric orbit, position, and proper motion, Available at: https://arxiv.org/abs/astro-ph/0002406 (Accessed: 8 September 2020).

9 thoughts on “The changing pole star”

  1. Hi Steve, what an interesting explanation. Thank you.
    Is the 25,800 year revolution a true circle or are there any other subtle forces at play which might distort it out of shape or perhaps make it elliptical, over the course of one or more revolutions?
    Regards, Roger

    Liked by 1 person

  2. Thanks Steve, very useful & informative.
    (One very minor quibble: on the south polar diagrams, the south ecliptic pole would be better idented as SEP rather than NEP)

    Liked by 2 people

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