Star Cluster Distances

Hyades

The Hyades cluster is also an open star cluster of around 200 stars in the constellation Taurus (White, J., p. 7). Located in the "nose" of the bull by the bright star Aldeberan, the Hyades has played a vital role in helping astronomers find the distances to other objects in the universe as well as estimate the size of the universe itself. Because the Hyades are located only around 150 light years from Earth, their trigonometric parallax angle could be easily measured, and the distance calculated (Hipparcos, online). Astronomers could then compare the H-R diagram main sequence of the Hyades to the main sequence of other clusters and estimate their distances from Earth (Brown, p. 21). This technique works great assuming the cluster members all move the same direction with the same velocity, and the distance to the Hyades is correct. Before the HIPPARCOS satellite was launched into orbit, a variety of studies by several different astronomers yeilded descrepancies in the actual distance to the Hyades, with values ranging from 135 to 180 light years (Brown, p. 17). This inconsistency cast doubt on every other measurement based on the Hyades distance.

While main sequence fitting can be helpful in determining distance approximations for star clusters, a far more accurate method involves measuring the parallax for each cluster. Parallax is the angular measurement of the apparent movement of a star in relation to background stars during a period of 6 months. Before the HIPPARCOS satellite was launched, the parallax angle for most stars was too small to detect. Recent parallax data gathered by the HIPPARCOS satellite has minimized the margin of error surrounding distance calculations, estimating the distance to the Hyades at around 150 light years, and allowing astronomers to more acurately estimate the true distance scale of the universe (Brown, p. 18).

Our Distance Calculations

Although the stars in a star cluster are all formed within the same gaseous cloud, they all have slightly different distances from Earth. The most accurate way to determine the distance to a star is to measure the parallax angle. As the earth moves around the sun, our view of a particular star will change with respect to the background stars. The angular measure of this movement is the parallax angle. From this angle, it is possible to calculate the distance to a star using the equation d = 1/p, where d is the distance in parsecs and p is the parallax angle in arcseconds.

The equation for distance is derived from trigonometry and the principle that relates arclength, radius, and angle measure of a circle, s = dq.

When applied to the Earth's position around the sun, the values of s and q can be substituted, and the equation rewritten as:
d = s/q.

 

 

s = dq
s/q = dq/q
d = s/q
substituting,
s = 1 AU
d = distance in parsecs
q = parallax angle in arcseconds

d = 1/ q

Using the parallax angles gathered by the HIPPARCOS satellite, we determined the approximate distances to ten stars of the pleiades. The average distance from Earth to the Pleiades is 112.6 parsecs or 367 light years.
 

From the distances of each star we calculated the B-V and apparent magnitude values (gathered from the HIPPARCOS data). We were also able to determine the absolute magnitude, and create an H-R diagram of these select stars. To determine the absolute magnitude, we used the equation m - M = 5 log d - 5. This equation is derived from the inverse-square law equation that relates apparent brightness to luminosity and distance, and the magnitude difference as it related to brightness ratio and the light detection capabilities of our eyes.

b = L / 4pd², where L is luminosity, b is apparent brightness, and d is distance to star in meters
m₂ - m₁ = 2.5 log (b₁/b₂), where m₂ and m₁ are apparent magnitudes, b₁ and b₂ are apparent brightnesses

m₂ - m₁ = 2.5 log ((L₁/4pd₁²)/(L₂/4pd₂²)
If  L₁ = L₂ then,
m₂ - m₁ = 2.5 log (d₂/d₁)²
m₂ - m₁ = 5 log (d₂/d₁)
m₂ - m₁ = 5 log d₂ - 5 log d₁

Absolute magnitude, M,  at distance of 10 pc,

m - M = 5 log d - 5,
where m is apparent magnitude, M is absolute magnitude, and d is distance in parsecs.

To view the table of data gathered for distance and absolute magnitude calculations for the Pleiades, click HERE

To view the H-R diagram created from the absolute magnitude calculations, click HERE.