Question

The apparent magnitude of a celestial object is how bright it appears from Earth. The absolute magnitude is its brightness as it would seem from a reference distance of 10 parsecs (pc). The difference between the apparent magnitude, \(m\), and the absolute magnitude, \(M,\) of a celestial object can be found using the equation \(m-M=5 \log d-5,\) where \(d\) is the distance to the celestial object, in parsecs. Sirius, the brightest star visible at night, has an apparent magnitude of -1.44 and an absolute magnitude of 1.45 a) How far is Sirius from Earth in parsecs? b) Given that \(1 \mathrm{pc}\) is approximately 3.26 light years, what is the distance in part a) in light years?

Short answer

a) 2.64 parsecs b) 8.60 light years

Step-by-step solution

Understand the Given Information

Identify the given values in the problem. For Sirius, we have the apparent magnitude \(m = -1.44\) and the absolute magnitude \(M = 1.45\). The equation that relates the apparent magnitude, absolute magnitude, and distance is given by \(m - M = 5 \log d - 5\).

Set Up the Equation

Substitute the given values into the equation \(m - M = 5 \log d - 5\). This yields \[ -1.44 - 1.45 = 5 \log d - 5. \]

Solve for \(\log d\)

Simplify the equation to isolate the logarithmic term: \[ -2.89 = 5 \log d - 5 \] Next, add 5 to both sides: \[ -2.89 + 5 = 5 \log d \] which simplifies to: \[ 2.11 = 5 \log d. \]

Isolate \(\log d\)

Divide both sides of the equation by 5 to solve for \(\log d\): \[ \log d = \frac{2.11}{5} = 0.422. \]

Solve for \(d\)

Exponentiate both sides using base 10 to find \(d\): \[ d = 10^{0.422} \approx 2.64 \text{ parsecs}. \]

Convert Parsecs to Light Years

Given that 1 parsec is approximately 3.26 light years, multiply the distance in parsecs by 3.26: \[ 2.64 \times 3.26 \approx 8.60 \text{ light years}. \]

Key concepts

absolute magnitude

Absolute magnitude is a measure of the true brightness of a celestial object. It represents how bright a star or other celestial object would appear if it were located exactly 10 parsecs (about 32.6 light-years) away from Earth. This helps astronomers compare the intrinsic brightness of objects without the distance influencing their perception. Imagine you have two light bulbs of the same power; one is close to you and the other very far away. You wouldn’t know they are equally bright just by looking at them because the one farther away appears dimmer. In astronomy, absolute magnitude corrects for this by standardizing the distance at which brightness is measured.

logarithmic equations

Logarithmic equations are used when we want to determine quantities that grow exponentially or need to invert exponential functions. In the context of astronomy, these come in handy for dealing with the vast ranges of brightness and distances. The equation used here, \(m - M = 5 \log d - 5\), is a logarithmic equation because it involves taking the logarithm of the distance d. Logarithms simplify the process of dealing with very large numbers by converting multiplicative relationships into additive ones. This makes calculations more manageable, especially in fields like astronomy where we deal with exceptionally large values.

distance conversion

Converting distances in astronomy often involves working with different units such as parsecs and light years. In the given problem, once we find the distance to Sirius in parsecs (approximately 2.64 parsecs), we use the conversion factor to translate this distance into light years. Converting parsecs to light years is straightforward: knowing that 1 parsec is approximately equal to 3.26 light years, we multiply the number of parsecs by 3.26. This yields the distance in light years, providing a more intuitive understanding for those more familiar with the light-year unit. For Sirius, this distance turned out to be approximately 8.60 light years.

parsecs to light years

Parsecs and light years are both units used to express astronomical distances, but they are based on different principles. One parsec is defined as the distance at which one astronomical unit subtends an angle of one arcsecond. Practically, this distance is about 3.26 light years. Light years measure the distance that light travels in one year, about 9.46 trillion kilometers or 5.88 trillion miles. To convert from parsecs to light years, simply multiply the number of parsecs by 3.26. This conversion is useful because light years can be easier to conceptualize for many people, providing a more tangible grasp on the immense distances in space.