The changing amplitude of an RR Lyrae star

Adapted from IOAA 2019 (Keszthely), theory problem 8. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

Some RR Lyrae stars show a slow, repeating change in the amplitude of their pulsation, called the Blazhko effect. An RR Lyrae star is observed at ${\lambda }_1=500$ nm and ${\lambda }_2=2000$ nm, which come from different layers of the star. Take the light at each wavelength as set by the radius and the blackbody intensity of its layer, and use Wien's approximation, $F(\lambda ,T)\propto {\lambda }^{-5}\exp (-C_{\text{b}}/(\lambda T))$, with $C_{\text{b}}=hc/k\approx 0.0144$ m K.

(a) The temperature of each layer varies between $T_1=6000$ K and $T_2=7400$ K. Ignoring changes of radius, find the ratio of the amplitudes in magnitudes at the two wavelengths. [5 points]

(b) Find the peak-to-peak amplitude at ${\lambda }_1$. [3 points]

(c) Ignoring changes of temperature, what does a change of radius between $0.9⟨R⟩$ and $1.05⟨R⟩$ add to the amplitude at any wavelength? [3 points]

(d) At the smallest amplitude of the Blazhko cycle the temperature varies only between 6100 K and 6900 K, while the radius changes little. Taking the temperatures of (a) for the largest amplitude, by how much does the amplitude change over the Blazhko cycle at each wavelength? [5 points]

(e) Which statements are true? There may be more than one. [4 points] (A) The Blazhko effect is easier to see in the infrared. (B) The changes of temperature dominate the visual light curve. (C) Ignoring changes of radius, the amplitude is inversely proportional to the wavelength. (D) Observations in several colours do not help in understanding the Blazhko effect.

Answer(a) 4. (b) About 1.0 magnitude. (c) 0.33 magnitudes. (d) About 0.4 magnitudes at 500 nm and 0.1 at 2000 nm. (e) B and C.