Question

The following passage is adapted from an essay on the history of early twentieth-century astrophysics and stellar measurement.

In the early twentieth century, astronomical research at the Harvard College Observatory relied heavily on 'computers'—talented women who painstakingly analyzed photographic glass plates of the night sky. Among them was Henrietta Swan Leavitt, who in 1902 began cataloging variable stars, whose brightness fluctuates over time. By 1908, Leavitt published an initial study of variable stars located within the Small Magellanic Cloud, a dwarf galaxy neighboring the Milky Way. Her investigation expanded in 1912 when she published a landmark paper examining twenty-five specific variable stars known as Cepheids within the same celestial cloud.

Leavitt recorded two distinct magnitude measurements for each Cepheid: its maximum brightness at the peak of its cycle and its minimum brightness at the trough. Because all twenty-five stars resided within the Small Magellanic Cloud, Leavitt reasoned that they were situated at roughly the same distance from Earth. Consequently, any observed difference in their apparent brightness reflected a true difference in their intrinsic luminosity rather than variations in distance. Upon plotting the stars’ brightness against their pulsation periods—the time required to complete one full cycle from maximum to minimum brightness and back—Leavitt discovered a striking relationship: brighter Cepheids possessed longer pulsation periods. Specifically, the logarithms of both variables exhibited a linear correlation.

To quantify this relationship, Leavitt selected two precise reference points along the smoothed curve of maximum brightness: a star with a period of 1.25 days corresponding to an apparent magnitude of 14.8, and a star with a period of 68 days corresponding to an apparent magnitude of 11.2. (In astronomical magnitude scales, smaller numerical values denote brighter celestial bodies.) She performed an identical linear alignment for the minimum brightness curve, noting that a period of 1.25 days yielded a minimum magnitude of 15.4, whereas a period of 68 days yielded a minimum magnitude of 12.5. By demonstrating that the period of a Cepheid directly indicated its intrinsic brightness, Leavitt established what would become known as the period-luminosity relation.

Leavitt’s discovery fundamentally altered astrophysics by providing a standard candle for cosmic distance measurements. Prior to her work, astronomers could measure distances only up to approximately 100 light-years using trigonometric parallax, a method relying on the shift in a star's apparent position as Earth orbits the Sun. Trigonometric parallax was useless for distant galaxies because the parallax angles were imperceptibly small. With Leavitt's period-luminosity relation, if an astronomer measured a Cepheid’s period of pulsation, its true intrinsic luminosity could be determined regardless of location. Comparing this intrinsic luminosity to the star’s apparent brightness observed through a telescope revealed its precise distance from Earth. In 1924, Edwin Hubble utilized Leavitt’s period-luminosity relation on Cepheids in the Andromeda Nebula, proving that Andromeda was an independent galaxy far beyond the Milky Way rather than a localized gas cloud, thereby expanding the known scale of the universe.

Based on the passage, when Leavitt evaluated the minimum brightness curve for the Cepheid variable stars in her 1912 study, a pulsation period of 68 days was directly associated with which apparent magnitude?