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We present the Apache Point Observatory BG40 broadband and simultaneousGemini $r$-band and $i$-band high-speed follow-up photometry observations andanalysis of the 40.5 minute period eclipsing detached double-degenerate binarySDSS J082239.54$+$304857.19. Our APO data spans over 318 days and includes 13primary eclipses, from which we precisely measure the system's orbital periodand improve the time of mid-eclipse measurement. We fit the light curves foreach filter individually and show that this system contains a low-mass DA whitedwarf with radius $R_A=0.031\pm0.006~{\rm R_\odot}$ and a$R_B=0.013\pm0.005~{\rm R_\odot}$ companion at an inclination of$i=87.7\pm0.2^\circ$. We use the best-fitting eclipsing light curve model toestimate the temperature of the secondary star as $T_{\rm eff}=5200\pm100~{\rmK}$. Finally, while we do not record significant offsets to the expected timeof mid-eclipse caused by the emission of gravitational waves with our 1-yearbaseline, we show that a $3\sigma$ significant measurement of the orbital decaydue to gravitational waves will be possible in 2023, at which point the eclipsewill occur about $8$ seconds earlier than expected.

Multi-band light curve analysis of the 40.5-minute period eclipsing double-degenerate binary SDSS J082239.54$+$304857.19