10 minute lesson · Mathematics · Original foundation material
Before you start: Fractions; Indices; Basic algebra.
Many astronomy calculations become shorter when you compare two situations before inserting large numbers. Powers describe how a quantity scales; logarithms turn a multiplicative change into an additive one.
Divide first, calculate second
If y = kxⁿ and k stays unchanged, dividing two cases gives y2/y1 = (x2/x1)ⁿ. The constant cancels. For y proportional to x², tripling x multiplies y by 9; for y proportional to 1/x², tripling x divides y by 9.
If several quantities change, keep each factor: y proportional to a²/b³ gives y2/y1 = (a2/a1)² ÷ (b2/b1)³. Write the ratios before doing the arithmetic.
A logarithm asks for an exponent
log10(1000) = 3 because 10³ = 1000. For positive a and b, log10(ab) = log10(a) + log10(b), and log10(a/b) = log10(a) − log10(b). A factor of 10 therefore adds 1 to a base-10 logarithm.
The logarithm of a measured quantity should use a dimensionless ratio, such as log10(F/F0), where F and its reference F0 have the same units. Do not replace log10(a+b) with log10(a)+log10(b).
Three bridges for broader mathematics
For equally likely outcomes, probability is the number of favourable outcomes divided by the total. Count ordered pairs separately when rolling two distinguishable dice.
A geometric series multiplies by the same factor each step. For |r| < 1, a + ar + ar² + … = a/(1−r). A derivative measures instantaneous rate: if position x(t) = at² + bt, then velocity dx/dt = 2at + b. These are starting points, not the complete NSEA mathematics syllabus.
A worked example
A detector has a dimensionless reading S = log10(F/F0). Its received intensity changes from 4F0 to 400F0. Find the change in S.
- The new intensity is 400/4 = 100 times the old intensity.
- The change in reading is log10(400) − log10(4).
- Apply the quotient rule: change = log10(400/4) = log10(100).
- Since 10² = 100, the reading increases by 2.
An increase of 2
Check your understanding
A quantity scales as R²/T. What is its new-to-old ratio if R triples and T doubles?
Show the explanation
3²/2 = 9/2 = 4.5. Apply each change to the correct power before combining them.